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Chapter 5. Macroweather and the climate

5.1 Expect, macroweather

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

this might have been the extended range 14 day weather forecast for Montreal on the 31 st 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 scalea. In that case (fig. 5.1b), we would 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 successive fluctuations.

The fact that macroweather fluctuations tend to cancel rather than accumulate – as in the weather regime, “wandering” up and down with prolonged increasing or decreasing swings is its defining feature. Quantitatively, it implies that the exponent H in time is negative rather than positive. Whereas with H>0, the weather is a metaphor for instability, in macroweather with H<0 there is systematic cancelation and the temperature appears to be stable. If we average over longer and longer times the variability is systematically reduced 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 macroweather in space?. As usual, we could explain the forecast with recourse to weather and macroweather maps. For example, fig. 5.2b (left column) shows the day to day evolution of the corresponding daily temperatures over the globe over the next four days. Focusing on Canada and the United States the (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 North American continent”.

While the appearance of the temperature maps for weather and macroweather 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 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

a In ch. 4, we saw that Mars was nearly such a macroweather climate with the transition at 1.8 sols.b In both weather and macroweather, 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 below.

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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), normalized them by their means (so that they all fluctuate around the value 1), and used a common vertical scale. By inspection, we can see that the macroweather time series is the exception with only small, nonintermittent fluctuations; indeed, the maximum is quite close to what would be expected if the process were Gaussian. On the contrary, in space (left column) macroweather is highly spiky as is the weather in both time and 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.

5.1.2 Macroweather and climate states, climate zonesThe strong intermittency of the weather regime is unsurprising and is due to its

turbulent nature discussed earlier. However, the averaging (here, over a month) to obtain the macroweather series greatly reduces (nearly eliminates) the temporal intermittency yet it completely fails to reduce the spatial intermittency: the intermittency of the spatial macroweather transect is even a bit stronger than 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 conditionsc, in order to highlight the relatively small month to month changes, fig. 5.2 showed “anomaly” maps. Just as the daily maps (fig. 5.2 left) defined anomalies as differences of the daily temperatures with the current one month average – the “macroweather state” - the macroweather series and maps (fig. 5.2, right column) are for anomalies obtained as the differences of the actual macroweather temperatures with the standard (World Meteorological Organization) thirty year average that implicitly defines “climate states”d.

Using macroweather states to define weather anomalies is a natural consequence of the weather-macroweather transition. Physically, weather anomalies correspond to the differences of the weather (daily here) with averages over several lifetimes of planetary structurese. However what is the justification for the ubiquitous use of monthly (macroweather) anomalies? Certainly it is convenient: whereas at one month f and at 1 - 2o

spatial resolution, over the globe, the anomalies typically vary in the range of a several degrees whereas the absolute reference temperature varies from one region to another by 70oC or more. 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.

c 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.d The slight additional complication is that for monthly resolution anomalies, one must remove the annual cycle. For example, officially ordained procedure for calculating 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.e As far as I know, such weather anomalies are not used in forecasting probably because the evolution of the atmosphere depends on the actual state of the atmosphere and not its difference with respect to the monthly average. While these weather anomalies are useful for highlighting the evolution of the atmospheric state from day to day, it does not help not to predict it.f 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.

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).

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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.

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).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.

But can the use of macroweather anomalies be justified objectively rather than just subjectively and 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 the thirty year duration was quietly kept and has not been questionned.

Fortunately, nearly eight decades after the thirty year period was originally adopted 1 an objective ex post facto justification was finally found, the basic evidence of which was already given in ch. 2, using both spectral and fluctuation analysis (figs. ?, ?). These

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analyses showed that - at least in the instrumental period (roughly 1850 - present g) that there is a new regime that starts at around 20- 30 years: the true climate regime. We saw that at longer times, the fluctuations increase rather than decrease, marking the end of macroweather and the beginning of the climate. Fig. 5.4 shows the transition time scale estimated directly from the 1871-2010 20CR reanalysis data at 2o resolution. Although the transition time varies somewhat with latitude, in the industrial epoch, a value of 30 years is a reasonable overall characterization. In later sections we discuss the anthropogenic origin of the current transition, and also the question of the transition in pre-industrial epochs.

In retrospect, it is hard to escape the conclusion that the original choice of a thirty climate normal duration was simply fortuitous, indeed, using preindustrial “multiproxies” described below, at the time, the climate was indeed stable with H<0 up to centennial scales or longer; the destabilizing anthropogenic warming was still too small to detect. For example in section 6.? (fig. ?) we find that at the time (1934) only about 0.3 oC of global anthropogenic warming had occurredh - barely above the natural variability (about 0.2 oC). In addition, over the entire period 1900-1930, the climate would have appeared to be quite stable with only about a 0.1oC increase due to anthropogenic effects, i.e. well below the typical year to year global temperature change due to purely natural causes i. To the IMO scientists, it would have been quite plausible that their climate normal period from 1900-1930 did indeed have a constant climate. Indeed, a simple estimate of the macroweather-climate transition scale is simply the time it takes for the anthropogenic warming to equal the natural variability (about 0.2oC per year), in 1934, this was about 60 years. Since then, emissions and other anthropogenic forcings have greatly increased, so that the time it takes for the anthropogenic warming to exceed the natural variability and to thus be evident of climate change has been reduced to only about 16 – 18 years. As a consequence, the 30 year weather-macroweather transition time (fig?, ?) is no more than an average over the recent epoch (since roughly 1880).

g 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 most climate applications.h From about 1750, the current warming is about 1 oC.i In the last four decades, it has required roughly 16- 18 years to for the warming to exceed the natural variability.

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Fig. 5.4: 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 j (top) as functions of latitudek. The thick curves show the mean over all the longitudes, and the dashed lines are the longitude to longitude variationsl. The macroweather regime is the regime between the top and bottom curves. Adapted from2.

Fig. 5.5: “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.

Now that we have justified the definition of anthropocene climate states as roughly thirty year averages, we can see what they look like. Fig. 5.5 shows the result using data over the 140 year period from 1871-2010. The data were divided into five non-overlapping 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. Finally, we can consider the climate state intermittency. Although there are not enough climate states (four or five at 28 year scales), to analyse the temporal intermittency, we can readily study it in space using the method of fig. 5.?, to determine the normalized absolute gradients (fig. 5.6). The figure shows that the intermittency is indeed very strong and that it is largely (but not only) due to coastlines and mountain ranges.

j Only valid in the anthropocene.k Estimated by the position of scale breaks in Haar structure functions from the 20CR reanalyse data. l The one standard deviation limits.

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Fig. 5.6: 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 45oN and 45oS. Note the different scales.

5.1.3 Characteristics of macroweatherBefore returning to the long time limit of macroweather - the transition to climate -

let us try to understand macroweather a bit better. Fig. 5.7 shows the spatial distribution of the key exponent H. Interestingly, the value of H is not constant as we move from one region to another, however, it is highly significant that it varies in the range from -1/2 to 0. The former corresponds to taking a sequence of independent random numbersm - a “white noise” - and the larger H’s up to H = 0 are obtained by increasingly smoothing such white white noises. Whereas white noise (H = -1/2) case has obviously no predictability, we will see in the chapter 6? that the closer H is zero, the more predictable the process is with the H = 0 limit in principlen having an infinite predictability. It is therefore significant that in the figure, we can see that the oceans tend to have a typical value H ≈-0.1, whereas land typically has H ≈-0.3: our ability to predict the air temperature over the oceans is therefore much higher than our ability to predict it over land.

In order to get an idea of what the different noises actually look like, fig. 5.8 shows examples of both real macroweather anomalies (right column), as well as simulations using a simple (nonintermittent) mathematical process known as fractional Gaussian noise (fGn, left hand column). At the bottom, (H = -1/2, white noise), there is a sequence of independent values with no relation between successive valueso. As one moves from

m Due to the “central limit theorem” in statistics, it doesn’t make too much difference how one determines the distribution of the random numbers, the key point is that the way that successive numbers are drawn should be identical and they should each be drawn independently.n Of course an infinite amount of past data would be needed to fully exploit it.o This statement is exactly true for the simulation, for the data it is only approximately true, being based on an estimate of H.

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bottom to top, H increases, and we see more and more correlations between successive values; at the top, coherent undulations are visible over most of the record. The basic 1680 point long series (corresponding to 140 years for the data) figure is shown in black; superposed is the same series but averaged over a factor of 12 in scale (blue; for the data, it is at annual resolution) and averaged and then rescaled (in red). We know from ch. 2(?) that when H<0, the effect of the averaging is to reduce the amplitude of the variations by a factor 12H; the rescaling was done by dividing each series by the factor 12 H to compensate for the averaging; as expected while the averaging without compensation (blue) can be very strong (for H = -1/2, bottom, it is reduction by a factor 121/2 = 3.46), the compensation does good job at restoring the range of variability so that it is comparable to the variability at the original one month resolutionp.

Although this strong resolution dependence is a basic feature of macroweather, at present, it neither adequately recognized nor taken into account either in GCM modelling, nor in empirical estimates of temperature. In the case of monthly and longer GCM series, the basic symptom is recognized: that the empirical and model temperatures tend to follow each other, but that the amplitude of GCM temperature variations are typically smaller than those of the empirical anomalies. Rather than recognizing the source of the problem as a resolution issueq – that the nominal GCM resolution is lower (it is more averaged) than the data - the problem is simply treated as yet another model imperfection to be corrected by ad hoc “post-processing”, now going by the sophisticated term “quantile matching”r. Unsurprisingly, the same resolution issue turns out to be important in empirical estimates of the earth’s temperature (see box 5.1 “How accurately do we know the temperature of the earth?”). This is because in order to make temperature maps on regular grids (e.g. 5 oX5o), the raw station and ship data are placed on the grid, spatially averaged over the grid box and then averaged over a month. Some grids have many measurements points, while some have few, and typically – since 1880 – about half have no data whatsoever s! While over half a dozen such instrumentally based data sets have been developed covering a century or more, each handles the variable and missing data issues differently and this results in different effective resolutions, none of which are exactly the same as the nominal resolution (1 month, 5oX5o). Since this resolution effect is multiplicative, it is important at all time scales and it ends up dominating the errors in estimates of decadal and centennial scale temperature changes that are needed notably for global warming4; box 5.? gives more details.

p The factors (12-H), top to bottom are: 1.28, 1.64, 2.11, 2.70, 3.46.q The resolution may have a “mismatch” in both space and in time.r Quantile matching is a general method of forcing the probability distribution of the data and the simulation to match. For Gaussian processes (a good approximation here), it is the same as multiplying the anomalies by the factor needed to match the standard deviation of the simulations and data, see e.g.:3 Zhao, T. et al. How Suitable is Quantile Mapping For Postprocessing GCM Precipitation Forecasts? J. Clim. 30, 3185-3196, doi:10.1175/JCLI-D-16-0652.1 (2017).s The fraction is somewhat variable, depending on the data set in question, but 50% is typical.

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Fig. 5.7: The spatial distribution of the fluctuation exponent H estimated from the NCEP

reanalysis; monthly anomaly data from 1948-2015 (?). Reproduced from Del Rio Amador et al 2017.

Fig. 5.8: A comparison of fractional Gaussian noise simulations (left) and selected twentieth century reanalysis anomalies (right) with the fluctuation exponent H increasing from - 1/2 to - 1/10 in steps of 1/10 (bottom to top)t. All are normalized by their absolute mean differences and are displayed using the same absolute axes. The plots are shown at two resolutions: high (black) and degraded by a factor of 12 (blue) and then – to statistically compensate for the cancellation - “renormalized” by multiplying by 12-H (red). For the 20CR, this corresponds to monthly and annual resolutions respectively.

t In addition to removing (usual) the thirty year average annual cycle we have also removed a linear trend which removes much of the global warming signal.

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Fig. 5.9a: Temperature anomalies (left to right) for the 1000 month period from 1926 (bottom) to 2010 (top) as a function of longitude (every 6o), from the 20CR, each series was offset in the vertical . Notice that the fluctuations with amplitudes that depend on position (bottom to top) and exponents H that also vary from position to position.

Fig. 5.9b: A stochastic macroweather simulation using complex cascades displaying fluctuations with amplitudes that depend on position (bottom to top) and exponents H that also vary from position to position.

A full understanding of macroweather requires knowledge not only of the spatial and temporal variability as a function of scale, but also of the joint space-time variability. Although this subject is still poorly studied, there are theoretical reasons (from cascade models), and from analysis of GCM outputs, to suppose that macroweather obeys a symmetry know as “space-time statistical factorization”. Such factorization means that different (spatially distributed) climate zones modulate the local temporal statistics without changing their type (e.g., their temporal scaling). The factorization principle is

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already implicitly used in practical climatology to “homogenize” the datau or to produce various climate indices that may be compared between different stations with different climates. With factorization, the temperatures at two possibly widely separated locations – such as between Montreal and the El Nino region off the coast of Equator – might be highly correlated with each other (“teleconnections”), yet the Equator temperature would not necessarily improve forecasts of Montreal temperatures. This is explained in detail ch. 6?, but by allowing forecasts at fixed locations to be made using only data at the given station (if the series is long enough in past!), it clearly allows for a great simplification.

5.2 Why macro “weather”?“Weather is the state of the atmosphere, to the degree that it is hot or cold, wet or

dry, calm or stormy, clear or cloudy. ...” (Wikipedia). If we combine this definition of weather with the usual view of the climate as average weather – “the climate is what you expect, the weather is what you get” - then there is no qualitative difference between them: the more that one averages the weather, the more climate-like the result becomes: in the long time limit, one finally obtains the climate. But if the weather-climate transition is simply a question of subjective averaging period, then how can the climate change?

Clearly without objective definitions, our understanding is limited and scientific progress is hindered. In retrospect, the discovery of the “synoptic maximum” – the drastic spectral transition at several days, first noticed by Van der Hoven in the 1950’s (section 4.?) – graphically misrepresented as a bumpv - and its subsequent relegation to a minor scalebound role, was a huge lost opportunity. We have seen that a scaling framework is required to tame the extraordinary atmospheric variability and that this shows that out to a hundred thousand years or so - ice age scales - there are three - not two regimes. There is no question that the high frequencies correspond to our common idea of weather, but what about the other regimes? If we insist that the climate immediately follow the weather then the low frequency regime past thirty years is something else: should we then call it the “climate change regime” or use the less cumbersome term “macroclimate”? Certainly this would lead to some bizarre usages, for example, the climate would change from month to month and current “long range” monthly predictions would be climate forecasts. When referring to periods thirty or more years in the past, we would talk about “past macroclimates”, ice cores and other paleo proxies would be “macroclimate indicators” and the fight against global warming would be a struggle to stop “macroclimate change”.

It would seem that to be consistent with common parlance, we should reserve the term “climate” for the third longer time regime. This leaves us with the middle regime: why call it “macroweather” rather than “microclimate”? To answer this, we have to consider how climate states can change, and both GCMs and stochastic cascade models help to understand this. GCMs are based on partial differential equations that embody the physical processes that are believed to govern the atmosphere, similarly, cascade models are based on the emergent higher level statistical laws. Bothw were originally weather models that

u For example, data from individual climate stations are normalized by using the station standard deviations or probability distributions.v Unfortunately, -as we saw in ch. 4 – the fact that it was a maximum and not simply a transition of spectral type was purely an artifact of the way that the graphs were plotted and this allowed scientists to think in scalebound terms of some localized disturbance rather than a change in character. w In the case of the “Fractionally Integrated Flux” (FIF, stochastic cascade) model, various extensions have been made (such as to couple them with an ocean model) as well as to include climate zones, but these models have no yet been much developed:5 Lovejoy, S. & de Lima, M. I. P. The joint space-time statistics of macroweather precipitation, space-time statistical factorization and macroweather models. Chaos 25, 075410, doi:10.1063/1.4927223. (2015).

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were later extended to include the oceans, the cryosphere, carbon cycle etc. and both can be used to objectively define the climates of models.

Let us focus on the more highly developed of the two modelling approaches, namely the GCMs. At some initial instant the state of the atmosphere (and in coupled models, the state of the ocean) is specified everywhere and then the model is “integrated” forward to determine the later states: mathematically, it is an “initial value problem”. Already, it was known from some theory – going back to Lorenz’s inverse cascade of forecast errors 6 in the 1960’s (discussed in ch. ?) and also by direct analysis of GCM outputs - that the errors in the large scale structures double every ten days or so. This error doubling time already provides a kind of “operational” definition of the weather: at scales longer than this, the results could no longer be interpreted deterministically because any small (microscopic) error would double in this time rapidly destroying the forecastx.

Although it is unrealistic, GCMs can be integrated forward from their initial conditions using fixed external conditions; notably with no variations in solar output, no changes in atmospheric composition (no changes in Greenhouse gases), no volcanism and no change in land use (e.g. deforestation). With everything external to the atmosphere fixed, the atmosphere displays “internal variability”, it varies in a quasi-steady manner around a long-term state, the model’s “climate”. In control runs one may thus directly observe the model’s convergence to its unique climate. In practice, this is done by taking centennial or millennial length simulations and by looking at long term averages or fluctuations around these averages. If one systematically studies the convergence using Haar fluctuations then - as expected due to the temporal scaling symmetry - one finds that the models have excellent scaling (fig. 5.9). Since H<0, these Haar fluctuations are essentially averages of the temperature anomalies with respect to the long term state so that the figure quantifies the rate of model convergence to its climate. The convergence was termed “ultra slow”7 since H ≈ -0.15y and from fig. 5.9, we can see that after 300 simulated years, the global temperature is still typically 0.1oC from its long term climatez. The small value of H implies that if the simulations were extended by a factor of 3000 that the resulting one million year averages would still be typically 0.03 oC from the true model climate!

In ch. 4, we already saw that GCMs had fairly realistic weather regime statistics; comparing fig. 5.9 with 2.? shows that the value H ≈ -0.15 is actually pretty close to the observed global macroweather temperature variations (i.e. up to about 20-30 years). GCM’s thus accurately reproduce the basic weather and macroweather statistics. Similarly, turbulent cascade models that were designed to reproduce the weather statistics (including multifractal intermittency), also reproduced the weather and macroweather, although with a somewhat more negativeaa H (fig. 5.9). Since these essentially weather models naturally produce macroweather at scales beyond the lifetimes of planetary structures, we can finally see that the term “macroweather” not “microclimate” is indeed appropriate.

x In Lorenz’ error doubling model, structures were predictable over their lifetimes; it is ironic that the connection was not made that the ten day predictability limit implied a ten day lifetime of planetary structures. yThis is the mean in fig. 5.9 but there is some variation from model to model.z The ultra slow convergence leads to technical difficulties in observing the climate state. This is because there is an initial “spin-up” period during which the initial state of the atmosphere - and especially oceans - are “spun-up” to adjust to their long term quasi-steady state. Although spin-up times of several (simulated) centuries are common, even this is not enough to prevent “model drift”; i.e. slow variations such as rising temperatures. In fig. 5.9, any drifts were removed by linear regression: only the residuals after removing a straight line were analyzed.aa More sophisticated models can reproduce different H values.

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5.3 Modelling the climate So what about climate change? From the point of view of the modellers then the only

way to change the model climate is by changing the solar output, by allowing volcanoes to erupt and by accounting for greenhouse gases and other human interventions: by changing the boundary conditions. In the words of Bryson8 “Climate is the thermodynamic/ hydrodynamic status of the global boundary conditions that 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. 9, 10) in which seasonal scale model validation is applied to climate scale predictions (for a recent discussion, see 11).

GCM modelling thus reduces climate change to a problem of changing boundaries, changing “climate forcings”, it ignores the possibility that some processes might have been forgotten. What if the model missed out on some slow decadal, centennial, millennial scale processes? If these decreased at long times, they might not prevent convergence to a climate state (i.e. H would still be <0), but they would imply that the model climate was different from the real one. However, if they were strong enough, increasing at long times, they could provide destabilizing internal sources of variability and cause the climate state to slowly vary, to “wander”bb.

Fig. 5.9 Top, red, the RMS Haar fluctuations for 11 CMIP5 control runscc. Control runs (top) actually 5500 months=458 years.Bottom: 2 multifractal simulations (2^17X2^5)= resolution, 1 day, 340 yearsTop slope = H =-0.15, bottom, -0.32 (multifractal simulations)

Bryson accurately described the interpretation of GCMs in terms of GCM climates, ultimately defining these in terms of the long time behaviour of control runs. But what about the real world? The clue of course is that since fixed conditions lead to pure macroweather (H<0), any long term deviations from this, with H>0 correspond to unstable variations of the long term averages, to changing climates. The climate regime is thus the regime in which climate states are themselves varying.

bb Of course, ultimately at very, very long scales such internal processes might themselves exhibit a converging new macroclimate regime.cc Selected for length and absence of over active El Nino (3- 5 year variability).

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We are now in a position to better understand the limitations and difficulties of climate modelling. Climate models are highly complex involving many assumptions about precipitation, radiation, ocean-atmosphere interactions and much more. Getting the climate “right” means that for a given set of boundary conditionsdd – climate “forcings” - we must realistically model the internal dynamics so that the ultimate state of sufficiently long enough control run averages - the model’s climate – corresponds to the real world climate.

In principle, to validate a climate model one must therefore run the model for a very long time and then compare the result to an earth-like planet with perfectly fixed forcings. Obviously, this is impossible,

In practice, one attempts to implement boundaries that are “realistic” and make them evolve according to historical data of past volcanic eruptions, past solar variations, past last use and past greenhouse gas levels, and then compare the result with the actual historical record, especially the temperature record.

Fig. 5.10 shows the result when this is done for a “Last Millenium” simulation using NASA’s E2-R model.

Fig. 5.10a: 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 7.

dd Pielke already criticizes this on the grounds that many of the boundaries such as atmosphere-land are not just passive but involve exchanges of energy and other fluxes:

12 Pielke, R. Climate prediction as an initial value problem. Bull. of the Amer. Meteor. Soc. 79, 2743-2746 (1998).

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Fig. 5.10a: 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 7.

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 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, 13).

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

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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!

1 Lovejoy, S. What is climate? EOS 94, (1), 1 January, p1-2 (2013).2 Lovejoy, S. & Schertzer, D. The Weather and Climate: Emergent Laws and

Multifractal Cascades. (Cambridge University Press, 2013).3 Zhao, T. et al. How Suitable is Quantile Mapping For Postprocessing GCM

Precipitation Forecasts? J. Clim. 30, 3185-3196, doi:10.1175/JCLI-D-16-0652.1 (2017).

4 Lovejoy, S. How accurately do we know the temperature of the surface of the earth? . Clim. Dyn., doi:doi:10.1007/s00382-017-3561-9 (2017).

5 Lovejoy, S. & de Lima, M. I. P. The joint space-time statistics of macroweather precipitation, space-time statistical factorization and macroweather models. Chaos 25, 075410, doi:10.1063/1.4927223. (2015).

6 Lorenz, E. N. The predictability of a flow which possesses many scales of motion. Tellus 21, 289–307 (1969).7 Lovejoy, S., Schertzer, D. & Varon, D. Do GCM’s predict the climate…. or

macroweather? Earth Syst. Dynam. 4, 1–16, doi:10.5194/esd-4-1-2013 (2013).

8 Bryson, R. A. The Paradigm of Climatology: An Essay. Bull. Amer. Meteor. Soc. 78, 450-456 (1997).

9 Palmer, T. N., Doblas-Reyes, F. J., Weisheimer, A. & Rodwell, M. J. Toward Seamless Prediction: Calibration of Climate Change Projections Using Seasonal Forecasts. . Bull. Amer. Meteor. Soc., 89, 459–470, doi:doi.org/10.1175/BAMS-89-4-459 (2008).

10 Palmer, T. N. Towards the probabilistic Earth-system simulator: a vision for the future of climate and weather prediction. Q.J.R. Meteorol. Soc. in press (2012).

11 Pielke, R. A. S. et al. in Complexity and Extreme Events in Geosciences (eds A. S. Sharma, A. Bunde, D. Baker, & V. P Dimri) (AGU, 2012).

12 Pielke, R. Climate prediction as an initial value problem. Bull. of the Amer. Meteor. Soc. 79, 2743-2746 (1998).

13 Lovejoy, S., Tuck, A. F., Hovde, S. J. & Schertzer, D. The vertical cascade structure of the atmosphere and multifractal drop sonde outages. J. Geophy. Res. 114, D07111, doi:07110.01029/02008JD010651. (2009).

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