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Chapter 5. Expect, macroweather

5.1 “Expect the cold weather to continue for the next ten days followed by a warm spell”;

this was the extended range 14 day weather for Montreal on the 31st of December, 2006, (fig. 5.1a). But imagine what it might have been if the earth rotated about its axis ten times more slowly; with the length of the day coinciding with the 10 day weather-macroweather transition time scale. In that case (fig. 5.1b), we might have heard: “expect mild weather on Monday, followed by freezing temperatures until a warm spell on Thursday followed by a brisk Friday and Saturday, a warming on Sunday and Monday followed by freezing on Tuesday, then a four day warm period followed by freezing and then warming…”. Whereas long term trends in weather can persist for up to ten days or more, in macroweather the upswings tend to be immediately followed by downswings (and visa versa) and longer term trends are much more subtle, being the result of imperfect cancellation of fluctuations.

The fact that a series of fluctuations tend to cancel rather than – as in the weather regime - “wandering” up and down with prolonged increasing or decreasing swings in either direction is the defining feature of macroweather. Quantitatively, it implies that the exponent H in time is negative rather than positive. Whereas the weather with H>0 is a metaphor for instability, in macroweather, due to the cancelling, the temperature appears to be stable, with averaging over longer and longer times reducing the variability so that it appears to converge to a well defined value, the “climate”. In more prosaic terms, the “macroweather is what you expect, the weather is what you get”.

But what about in space? Fig. 5.2 left (weather) and right (macroweather) appear somewhat different1. As usual, we could explain the forecast with recourse to weather and macroweather maps. For example, fig. 5.2 (left) shows the day to day evolution of the corresponding daily temperatures over the globe over the next four days. Focusing on the North American continent (within the green ellipses), we would have been told that “a mass of warm air will be gradually displaced by colder arctic air descending from the north west, covering the continent by Thursday.” In the macroweather planet (right column fig, 5.2), the explanation might be: “The mass of unusually cold air currently over the continent will shrink on Tuesday, spread to the northeast on Wednesday and by Thursday will expand covering most of Canada and the United States”.

While the appearance of the temperature maps for weather (Fig. 5.2 left) and macroweather (right) appear to be a bit different, it turns out that they both have fairly similar, smooth behaviour in space (positive spatial H’s with comparable values) so that they are mostly distinguished by the way that they evolve in time: the sign of the temporal H. But H only characterizes typical, average fluctuations; recall that in ch. 1 we saw how a fairly innocent looking aircraft transect hid very strong variability, “spikiness”, “intermittency”. To bring this out, consider fig. 5.3 that compares the spikiness of weather and macroweather, both in space (bottom row) and in time (top row). To make the comparisons as fair as possible, we have presented 360 points for each (corresponding to a spatial resolution of 1o longitude and 1 hour, one month in time). Following fig. 1?, we have taken the absolute differences (so that the minimum is zero), then normalized them by their means (so that they all fluctuate around the value 1), finally we used a common vertical scale. By inspection, we can see that the macroweather time series is the odd one out with

1 In both weather and macroweather case, in order to bring out the temperature changes which are relatively small with respect to the absolute temperatures, we have used anomalies, in the first case with respect to the average for the whole month, in the second for the previous thirty Januaries; see the text.

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only small, nonintermittent fluctuations; indeed, the maximum is quite close to what would be expected if the process were Gaussian. On the contrary, in both time and in space (left column), the weather is highly spiky as is macroweather in space. Indeed, if any of these three were produced by a Gaussian process, their maxima would correspond to probabilities of less than one in a trillion.

The strong intermittency of the weather regime is unsurprising and is due to its turbulent nature discussed earlier. Similarly, the averaging (here, over a month) to obtain the macroweather series greatly reduces (nearly eliminates) the temporal intermittency. However, it completely fails to reduce the spatial intermittency: the intermittency of the spatial macroweather transect actually quantitatively a bit stronger than the weather regime intermittency! This turns out to be the statistical consequence of the existence of “climate zones”: the fact that huge spatial variability persists over long periods of time characterizing fairly stable “climate states”. Indeed, due to this long term persistence of stable atmospheric conditions2, the maps shown are for “anomalies”; not for the temperatures themselves and the use of anomalies highlights the relatively small changes. Just as the daily maps (fig. 5.2 left) defined anomalies as differences of the daily temperatures with the current “macroweather state” (one month average), the macroweather series and maps are for anomalies obtained as the differences of the actual macroweather temperatures with the deseasonalized thirty year averages that thus implicitly define “climate states”3.

The idea of macroweather states and using them to define weather anomalies is a natural consequence of the weather-macroweather transition and physically it corresponds to averages over several lifetimes of planetary structures. But what about climate states and the justification for effectively using them to define the macroweather anomalies in fig. 5.2? Although it is nearly always used, this macroweather anomaly definition is purely conventional - being codified by the World Meteorological Organisation. For example, the January anomalies used in fig. 5.2 are the differences between the January monthly averaged temperature and the average of all the Januaries over the previous thirty year reference period (by convention updated every decade), similarly for the other months of the year. Whereas at one month4 and at 1 - 2o spatial resolution, the anomalies typically vary in the range of a few degrees whereas over the globe the absolute reference temperature may vary by 70oC from one region to another. Had we shown the monthly variation of the actual temperatures rather than the anomalies, we wouldn’t have seen much beyond the seasonal temperature variation.

2 By definition, the atmosphere is stable in the macroweather regime, what is not so obvious is that each different region has significantly different averages, characteristics.3 The slight additional complication is that for monthly resolution anomalies, one must remove the annual cycle as discussed below.4 Recall that since in macroweather H<0 (in time) therefore averaging to longer times – such as one year – will reduce the amplitudes of the anomalies by 12H≈ 0.60 for the typical value H =-0.2.

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Fig. 5.1a: The mean daily temperatures in Montreal, Canada for Jan. 1- 14, 2006.

Fig. 5.1b: Macroweather temperatures for Montreal obtained by rescaling Montreal macroweather temperature anomalies from monthly resolution data from January 2000 through February 2001. The mean (-1 oC) was adjusted to be the same as in fig 5.1a and it was scaled so that the spread about the mean (the standard deviation, 4.9 oC) was also the same.

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Fig. 5.2: Left column: Average daily temperatures for January 1 - 4 (top to bottom) from the ECMWF reanalysis for the month of January 2006 (used in fig. 5.1a), at 1.5o spatial resolution. To bring out the small changes, the anomaly with respect to the overall January average temperature is shown. The data are from ±60o latitude (this avoids much of the map projection distortion).

Right: Average monthly temperatures for the twentieth century reanalysis for the first four months of 2000 (used in fig. 5.1b), at 2o spatial resolution. To bring out the small changes, the anomaly with respect to the average temperature for the previous 30 Januaries is shown.

Blue indicates negative anomalies, red positive anomalies the green circle around North American is discussed in the text.

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Fig. 5.3: “Climate states” using the 28 year period 1871-1898 as the reference, data from the 20CR, ±60o latitude. Blue shows little change, red shows much change (increase) in temperatures. The regions most sensitive to global warming are the most red.

Why 30 years? In chapter 1, we traced its origin to the original “climate normal” defined by the International Meteorological Organization as the period from 1900-1930. As it became clear that the climate was changing, the reference period was changed - at first every thirty years, now every decade - but keeping without scientific justification the thirty year duration. Then, in ch. 2, using spectral and fluctuation analysis (figs. ?, ?), we saw that - at least in the instrumental period (roughly 1850 - present5) and for globally averaged temperatures, that there is a new regime at around 20- 30 years. Using fluctuation analysis (fig. ?) we saw that at longer times, the fluctuations started to increase again marking the end of the macroweather regime and the beginning of the climate regime. Fig. 5.4 shows the transition time scale estimated directly from the 1871-2010 20CR reanalysis data at 2o

resolution. Although the transition varies somewhat with latitude, at least in the industrial epoch, a value of 30 years is a reasonable overall characterization.

We shall see later that even by 1930, only about 0.3oC of global anthropogenic warming had occurred6 - barely above the natural variability (about 0.2 oC) - and the period 1900-1930 would have witnessed only about a 0.1oC change i.e. well below the “signal to noise” detection level7. Since then, emissions and other anthropogenic forcings have increased, so that over the last four decades, the same global change (0.1 oC) occurs every 8

5 As discussed in ch. 1,? instrumental temperatures measurements existed well before 1850, but it was only in the second half of the 19th century that they became sufficiently numerous for many purposes.6 From about 1750, the current warming is about 1 oC.

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- 9 years requiring about double length of time for the warming to exceed the natural variability. In other words, the weather-macroweather transition time is decreasing on the figure 30 years is an average over the entire period since about 1880. In retrospect it is thus fortuitous that the original 30 year climate normal duration coincides so well with this epoch averaged transition time scale. Beyond this tendency for the time scale to be reduced as we approach the present, there is also some geographical variability. Fig. 5.3 shows the result using the 140 years period from 1871-2010 using twentieth century reanalysis data.

Again, in order to bring out the small changes (of the order of a degree or some), the data were divided into five nonoverlapping 28 year periods (nearly 30 years) and the differences with respect to the reference 28 year period (1871-1898) are shown. Unsurprisingly, the figure mostly displays a fairly uniform warming trend that we will show is primarily due to human activities.

Fig. 5.?: The variation of the weather-macroweather transition scale (bottom, an extract of the curves in fig. 4.10 where it is described) and the macroweather - climate transition scale 8 (top) as functions of latitude9. The thick curves show the mean over all the longitudes, and the dashed lines are the longitude to longitude variations10. The macroweather regime is the regime between the top and bottom curves. Adapted from1.

that corresponds also to industrialization and increased anthropogenic influences – at first changes in land use and then increasingly fossil fuels

7 In the last four decades, it has required roughly 16- 18 years to for the warming to exceed the natural variability. 8 Only valid in the anthropocene.9 Estimated by the position of scale breaks in Haar structure functions from the 20CR reanalyse data. 10 The one standard deviation limits.

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If we lived on a more Mars like planet with weather macroweather transition at about one day instead of 5- 10 days,

“The extended range forecast calls for mild weather Monday, followed by freezing temperatures until a warm spell Thursday followed by cold Friday and Saturday, warming on Sunday and Monday followed by freezing on Tuesday, then a four day warm period followed by freezing and warming…”

Weather: “The extended range forecast calls for cold weather to continue for the next ten days followed by a warm spell

Fig.5.3 The spikiness” (intermittency) in time and space of weather and macroweather series, compared. The graphs show absolute differences of east-west spatial transects (bottom) and series (time, top) at weather scales, (left), and macroweather scales (right). The graphs are each from 360 points (in space, at 1o resolution), and show the absolute differences between consecutive values. All the series were normalized by their means. While the spatial intermittencies (bottom) are not too different (at macroweather scales, it is a bit stronger), the temporal intermittencies are totally different, nearly absent from the 4 month (macroweather) series (upper right).

Upper left: Hourly temperature data from January 1 -15, 2006 from a station in Lander Wyoming. The maximum value is 8.23 standard deviations above mean, the process is highly non Gaussian (for a Gaussian process with 360 points, the maximum would be at roughly 2.8 standard deviations above the mean).

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Upper right: 20CR reanalysis from 1891-2011, each point is a four month average, the data are for a 2ox2o grid point over from Montreal, Canada (45oN). The maximum is 2.53 standard deviations above the mean, close to that of a Gaussian (for a Gaussian process with 360 points, the maximum would be at roughly 2.8 standard deviations above the mean).

Lower left: ECMWF reanalysis for the average temperature of 21st January, 2000, at 45oN; the maximum value is 7.26 standard deviations above mean, the process is highly non Gaussian.

Lower right: ECMWF reanalysis: the monthly averaged temperature for January 2000 at 45oN. The maximum is 7.66 standard deviations above the mean.

Fig. 5.? The east-west absolute gradients of the temperature climate state obtained by averaging over 140 years from 1871 to 2010. The data are between ±60o latitude and were taken from the twentieth century reanalysis at 2o spatial resolution. For each latitude, the gradients are normalized by the mean gradient at that latitude. Left: The gradients from successive latitudes are offset by 2 units in the vertical; one can roughly make out the, major mountain ranges and coastlines. Right: specific examples at 45 oN and 45 oS. Note different scales.

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Fig. 5.? RMS: 11 control runsControl runs (top) actually 5500 months=458 years.2 multifractal simulations (2^17X2^5)= resolution, 1 day, 340 yearsTop slope=-0.15, bottom, -0.32 (muotifractal simulations)

( ranges normalized (seperately for time and space), note that the lower left is the 21 Jan, 2000 ECMWF, Lander, hourly: 8.23 sd' s above the mean20CR four month averages:, max is 2.53 sd’s above the mean1/360 corresponds to 2.78 sd' sEcmwf, jan. 2000 in space max is 7.6606842 sd' s above mean

Meteorological and climatological sciences have become increasingly distinct yet there is still no generally accepted definition of the climate – or what is nearly the same thing – what precisely is the distinction between the weather and the climate? And if our notion of the climate is vague, what do we mean by climate change?

While atmospheric scientists routinely use the expressions “climate scales” and “meteorological scales” the actual boundary between them is not clear and most improve little upon the dictum:

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“The climate is what you expect, the weather is what you get”. -Farmer’s Almanac

This is actually quite close to the principal definition given by the US National Academy of Science:

“Climate is conventionally defined as the long-term statistics of the weather...”

-2.

Which improves on the Almanac only a little by proposing:

“…to expand the definition of climate to encompass the oceanic and terrestrial spheres as well as chemical components of the atmosphere”.

Another official source attempts to quantify the issue of the exact meaning of “long term” by tentatively suggesting a month as the basic “inner” climate time scale… but ultimately it seems to yield to a higher authority:

“Climate in a narrow sense is usually defined as the "average weather," or more rigorously, as the statistical description in terms of the mean and variability of relevant quantities over a period of time ranging from months to thousands or millions of years. The classical period is 30 years, as defined by the World Meteorological Organization (WMO). These quantities are most often surface variables such as temperature, precipitation, and wind. Climate in a wider sense is the state, including a statistical description, of the climate system.”

-Intergovernmental Panel on Climate Change AR4, Appendix I: Glossary, p. 942, 3.

At first sight, the last sentence is an interesting addition but is ultimately tautological since it defines the climate as the statistics of the “climate system”, which itself is left undefined.

Finally, an attempt at a more comprehensive definition is only a little better:

Climate encompasses the statistics of temperature, humidity, atmospheric pressure, wind, rainfall, atmospheric particle count and numerous other meteorological elements in a given region over long periods of time. Climate can be contrasted to weather, which is the present condition of these same elements over periods up to two weeks... Climate … is commonly defined as the weather averaged over a long period of time. The standard averaging period is 30 years, but other periods may be used depending on the purpose.also includes statistics other than the average, such as the magnitudes of day-to-day or year-to-year variations… The difference between climate and weather is usefully summarized by the popular phrase "Climate is what you expect, weather is what you get."

-Wikipedia

What is new here is the explicit attempt to distinguish weather (periods less than two weeks) and climate (30 years or more). However, as with the IPCC definition, these time

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periods are simply suggestions, with no attempt at physical justification. In any case they leave the intervening factor of 1000 or so in scale (literally!) up in the air.

An obvious problem with these definitions is that they fundamentally depend on subjectively defined averaging scales. This fuzziness is also reflected in numerical climate modelling since Global Climate Models (GCM’s) are fundamentally the same as weather models but at lower resolutions, with a different assortment of subgrid parametrisations and they are coupled to ocean models and – increasingly – to carbon cycle, cryosphere and land use models. Consequently, whether we define the climate as the long-term statistics of the weather, or in terms of the long-term interactions of components of the “climate system”, we still need an objective way to distinguish it from the weather. These problems are clearly compounded when we attempt to objectively define climate change.

However, there is yet another difficulty with this and allied definitions: they imply that climate dynamics are nothing new: that they are simply weather dynamics at long time scales. This seems naïve since we know from numerous examples in physics that when processes repeat over wide enough ranges of space or time scale they typically display qualitatively new features so that over long enough time scales we expect that new climate laws should emerge from the higher frequency weather laws. These qualitatively new emergent laws could simply be the consequences of long range statistical correlations in the weather physics in conjunction with qualitatively new climate processes – due to either internal dynamics or to (external) orbital, solar, volcanic or anthropogenic forcings - their nonlinear synergy giving rise to emergent laws of climate dynamics.

A useful definition of climate should involve a physical basis for the distinction/boundary between weather and climate as well as an identification of each regime with specific mechanisms and a corresponding specific type of variability. Both of these ingredients are provided by the scaling approach. In this framework, weather processes are those whose space-time variability follow the emergent turbulent laws discussed earlier. We have seen in ch. 8 that the driving solar energy flux combined with the finite size of the earth implies a drastic “dimensional transition” in the behaviour at the lifetime of the corresponding atmospheric structures at tw ≈ 10 days and for ocean structures at to ≈ 1 yr. Both time scales not only follow directly from the energy flux based theory of the horizontal variability/dynamics (and tw, directly from realistic estimates of the solar forcing), but – at least in the case of tw - also correspond to the observed sharp change in the scaling of all the atmospheric fields somewhere in the 5 -20 day (“synoptic maximum”) region, followed by a flatter “plateau” low frequency weather regime. At least for the temperature, over time scales between tw and to, the ocean boundary condition has strongly turbulent variability with C1 ≈ 0.1, a ≈1.8, H ≈0.5 so that its effect on atmospheric buoyancy and humidity is to increase the intermittency of monthly average temperatures over the land and ocean (c.f. fig. 8.6 e).

Since the atmosphere is a nonlinear dynamical system with interactions and variability occurring over huge ranges of space and time scales, the natural approach is to consider it as a hierarchy of processes each with wide range scaling, i.e. each with nonlinear mechanisms that repeat scale after scale over potentially wide ranges, (see fig. 1.9 c, d). Following 4, Schmitt et al., 1995, Pelletier, 1998, 5, 6, 7, 8-10 this approach is increasingly superceding earlier approaches that postulated more or less white noise backgrounds with a large number of spectral “spikes” corresponding to many different quasi-periodic processes. This includes the slightly more sophisticated variant 11 which retains the spikes but replaces the white noise with a hierarchy of Ornstein-Uhlenbeck processes (white noises and their integrals: in the spectrum, “spikes” and “shelves”; see fig. 8.6 d). In appendix 10B we discuss the corresponding stochastic linear forcing approaches. Finally

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we could also mention 12 who proposed a “hybrid” composite which includes a single short range scaling regime.

The resulting “three scaling regime” weather- low frequency weather – climate model was already empirically illustrated in fig. 1.9 c. The label “weather” for the high frequency regime seems obvious and requires no further comment. Similarly the lowest frequencies correspond to our usual ideas of multidecadal, multicentennial, multimillennial variability as “climate”. However labelling the intermediate region “ low frequency weather” - rather than say “high frequency climate” – needs some justification. The point is perhaps made more clearly with the help of fig. 1.9 d which shows a blowup of fig. 1.9 c with both global and locally averaged instrumentally based spectra as well the corresponding spectra from a GCM control run (see the caption for details and sections 11.3.3, 11.3.4). While the spectrum of the data (especially those globally averaged) begin to rise for frequencies below ≈ (10 yrs)-1, the control run spectrum maintains its relatively flat “plateau” – like behaviour out to at least (500 yrs)-1. Similar conclusions for the control runs of other GCMs at even lower frequencies were found by 13, 14, 15 and 10 (see section 11.3.3 for various GCM simulations and a detailed discussion) so that it seems that in the absence of external climate forcing, the GCM’s – which are thus essentially low frequency weather models – reproduce the plateau but not the lower frequency climate regime with its characteristic strong spectral rise. Similarly, we shall see in the next section that when the FIF cascade models that reproduce the weather scale statistics are extended to low frequencies they too predict the “plateau”, and they too need some new ingredient to yield the lower frequency climate regime. In this chapter, we therefore focus on low frequency weather, the climate regime proper is the subject of ch. 11.

In his monumental “Climate: Past, Present, and Future” Horace Lamb argued that the early scientific view was “climate as constant” 16. Reflecting this, in 1935 the International Meteorological Organization adopted 1901-1930 as the “climatic normal period”. Following the post war cooling, this view evolved: for example the official American Meteorological Society glossary 17 defined the climate as “the synthesis of the weather” and then “…the climate of a specified area is represented by the statistical collective of its weather conditions during a specified interval of time (usually several decades)”. Although this new definition in principle allows for climate change, the period 1931-1960 soon became the new “normal”, the ad hoc 30 year duration became entrenched, today 1961-1990 is commonly used. Mindful of the extremes, Lamb warned against reducing the climate to just “average weather”, while viewing the climate as “…the sum total of the weather experienced at a place in the course of the year and over the years”, 16.

Lamb’s essentially modern view allows for the possibility of climate change and is closely captured by the popular expression: “The climate is what you expect, the weather is what you get” (the character Lazurus Long in 18, but often attributed to Mark Twain). It is also close to the US National Academy of Science definition: “Climate is conventionally defined as the long-term statistics of the weather…” 2 which improves on the “the climate is what you expect” idea only a little by proposing: “…to expand the definition of climate to encompass the oceanic and terrestrial spheres as well as chemical components of the atmosphere”.

The Twain/Heinlein expression was strongly endorsed by the late E. Lorenz who stated: “Before embarking on a search for an ideal definition (of climate) assuming one exists, let me express my conviction that such a definition, when found must agree in spirit with the statement, “climate is what you expect”.” 19. He then proposed several definitions based on dynamical systems theory and strange attractors (see also 20).

A variant on this, motivated by GCM modeling, was proposed by 21 (criticized by 22): “Climate is the thermodynamic/ hydrodynamic status of the global boundary conditions that

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determine the concurrent array of weather patterns.” He explains that whereas “weather forecasting is usually treated as an initial value problem … climatology deals primarily with a boundary condition problem and the patterns and climate devolving there from”. This definition could be paraphrased “for given boundary conditions, the climate is what you expect”. This and similar views provide the underpinnings for much of current climate prediction, including the recent idea of “seamless forecasting” (e.g. 23, 24) in which seasonal scale model validation is applied to climate scale predictions (for a recent discussion, see 25).

There are two basic problems with the Twain/Heinlein dictum and its variants. The first is that they are based on an abstract weather - climate dichotomy, they are not informed by empirical evidence. The glaring question of how long is “long” is either decided subjectively or taken by fiat as the WMO’s “normal” 30 year period. The second problem – that will be evident momentarily – is that it assumes that the climate is nothing more than the long-term statistics of weather. If we accept the usual interpretation – that we “expect” averages - then the dictum means that averaging weather over long enough time periods converges to the climate. With regards to external forcings, one could argue that this notion could still implicitly include the atmospheric response i.e. with averages converging to slowly varying “responses”. However, it implausibly excludes the appearance of any new “slow”, internal climate processes leading to fluctuations growing rather than diminishing with scale.

To overcome this objection, one might adopt a more abstract interpretation of what we “expect”. For example if the climate is defined as the probability distribution of weather states, then all we “expect” is a random sample. However even this works only inasmuch as it is possible to merge fast weather processes with slow climate processes into a single process with a well defined probability distribution. While this may satisfy the theoretician, it is unlikely to impress the layman. This is particularly true since to be realistic we will see that one of the regimes in this composite model must have the property that fluctuations converge whereas in the other, they diverge with scale. To use the single term “climate” to encompass the two opposed regimes therefore seems at best unhelpful and at worst misleadling.

The main purpose of this paper is show that the weather-climate dichotomy is empirically untenable, that hiding between the two is a missing middle range spanning a factor of a thousand in scale (≈ 10 days to ≈ 30 years) characterized by qualitatively different dynamics. This new low frequency weather regime was dubbed “macroweather” since it is a kind of large scale weather (not small scale climate) regime 26, it fundamentally clarifies the distinction between weather and climate, the status and role of GCM models and the notion of climate predictability.

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Fig. 5c: Haar fluctuation analysis of globally, annually averaged outputs of past Millenium simulations over the pre-industrial period (1500-1900) using the NASA GISS E2R model with various forcing reconstructions. Also shown (thick, black) are the fluctuations of the pre-industrial multiproxies showing that they have stronger multi centennial variability. Finally, (bottom, thin black), are the results of the control run (no forcings), showing that macroweather (slope<0) continues to millennial scales. Reproduced from 27.

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Fig. 5d: Haar fluctuation analysis of Climate Research Unit (CRU, HadCRUtemp3 temperature fluctuations), and globally, annually averaged outputs of past Millenium simulations over the same period (1880-2008) using the NASA GISS E2R model with various forcing reconstructions (dashed). Also shown are the fluctuations of the pre-industrial multiproxies showing the much smaller centennial and millennial scale variability that holds in the pre-industrial epoch. Reproduced from 27.

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

Readers of the blog “23/9 D atmospheric motions: an unwitting constraint on Numerical Weather Models” will recall that larger and larger atmospheric structures become flatter and flatter at larger and larger scales, but that they do so in a scaling (power law) way. Contrary to the postulates of the classical 3D/2D model of isotropic turbulence, there is no drastic scale transition in the atmosphere’s statistics. However, since the famous Global Atmospheric Sampling Program (GASP) experiment (fig. 2) there have been repeated reports of drastic transitions in aircraft statistics (spectra) of horizontal wind typically at scales of several hundred kilometers. We are now in a position to resolve the apparent contradiction between scaling 23/9D dynamics and observations with broken scaling. At some critical scale – that depends on the aircraft characteristics as well as the turbulent state of the atmosphere - the aircraft “wanders” sufficiently off level so that the wind it measures changes more due to the level change than to the horizontal displacement of the aircraft. It turns out that this effect can easily explain the observations. Rather than a transition from characteristic isotropic 3D to isotropic 2D behavior (spectra with transitions from k-5/3 to k-3 where k is a wavenumber, an inverse distance), instead, one has

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a transition from k-5/3 (small scales) to k-2.4 at larger scales (fig. 2), the latter being the typical exponent found in the vertical direction (for example by dropsondes, 28).

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

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

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