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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 scale, an alignment of scales that is almost achieved on Marsa. 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 tendency of macroweather fluctuations to cancel rather than accumulate is its defining feature, the systematic cancelation means that it is stable. Quantitatively, it implies that the temporal fluctuation exponent H is negative. In the weather regime with positive H, the temperature, wind and other variables wander up and down with prolonged swings: the weather is a metaphor for instability. If we average macroweather over longer and longer times its variability is systematically reduced so that it appears to converge to a well defined value: “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) shows the day to day evolution of the daily temperatures corresponding to the forecast in fig. 5.1a.. Focusing on Canada and the United States (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 (fig. 5.2, right): “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 America”.

While the appearance individual temperature maps for weather and macroweather are not so different, they are both fairly smooth in space (positive spatial H’s with comparable values); mainly distinguished by the way that they evolve in time: the signs of their temporal H’s. But H only characterizes typical, average fluctuations; 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 of longitude and 1 hour and one month in time). Following fig. 1?, we have taken the absolute differences (so that the minimum is zero), divided them by

a In ch. 4, we saw that Mars was nearly such a macroweather world 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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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 in ch. 4. However, the averaging 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 climate states that change very slowly. Of course, at each location, the long term temperature characteristics are combined with long term relationships (correlations) with other variables, notably precipitation - to yield the familiar physical geography climates: “Mediterranean”, “temperate”, “dessert” etc. In order to highlight the relatively small month to month changes with respect to these longer term atmospheric conditionsc, the maps in fig. 5.2 showed “anomalies”, not absolute temperatures. Just as the daily maps (fig. 5.2 left) defined anomalies as differences of the daily temperatures with the current one month average – the current “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 the current climate stated (“climate normal”).

There are thus two averaging periods that are conventionally used: one month and thirty years, neither have been justified on objective scientific grounds. For example in climate science, the use of monthly averages is ubiquitous, but on the rare occasions when it is justified at all, it is only by its convenience: although a month is not even a precise duration, it is well adapted to human systems such as national weather servicese. Fortunately, a duration of one month is quite close to the weather-macroweather transition time scale, so that it can be (retrospectively) theoretically justified. Monthly averages effectively define “macroweather states”; physically these are averages over several lifetimes of planetary structures. These states can then be used to define weather anomaliesf as the differences of the weather (itself defined using weather scale averages,

c By definition, the atmosphere is stable in the macroweather regime; what is not so obvious is that each different region has significantly different characteristics.d The slight additional complication is that for monthly resolution anomalies, one must remove the annual cycle. For example, the 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 An informal mini-survey of colleagues revealed that most accepted monthly resolutions as convenient. At best one opined that averaging over a month “reduces noise” but this only makes sense if one assumes that there is a “weather noise” that is objectively distinct from a macroweather “signal”, so monthly averaging still requires a scale break to be justified.f As far as I know, such weather anomalies are not used, probably because the evolution of the atmosphere depends on the actual state of the atmosphere and not its difference with respect to the monthly (macroweather) average. While these weather anomalies are useful for highlighting the day to day evolution of the atmospheric state from, it does not help not to predict it.

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here, we average over a dayg). However what is the justification for the ubiquitous use of monthly (macroweather) anomalies defined with respect to 30 year averages? Certainly it is convenient: whereas at one monthh 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 fig. 5.2 shown the monthly variation of the actual temperatures rather than the anomalies, we wouldn’t have seen much beyond the seasonal temperature variation.

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

Fig. 5.1b: Macroweather temperatures for Montreal obtained by from 14 consecutive monthly anomalies (denoted “macrodays” for this fictitious planet) 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.

g Ignoring the diurnal cycle, in the weather regime, H>0 so that weather anomalies will not be sensitive to this averaging period.h 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. 5.2: 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.Right: Average monthly temperatures for the 20C reanalysis for the first four months of 2000 (used in fig. 5.1b), at 2o spatial resolution. To bring out the small changes, the anomalies with respect to the average temperature over the previous 30 Januaries, 30 Februaries, etc. are shown.

Blue indicates negative anomalies, red positive anomalies the green circle around North American region 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 time series (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, the temporal intermittencies are nearly absent from the 4 month averaged (macroweather) series (upper right).

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

Lower left: ECMWF reanalysis for the average temperature of 21st January, 2000, along the 45oN parellel at resolution of 1o longitude; the maximum value is 7.26 standard deviations above mean. Lower right: The same as at left but for the temperature averaged over monthly January 2000. 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 the origin of this time scale to the rather arbitrary “climate normal” defined by the International Meteorological Organization as the period from 1900-1930. When it became clear that the climate was changing, the reference period was regularly updated - at first every thirty years, now it is every decade - but its thirty year duration has quietly been kept.

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 analyses showed that - at least in the instrumental period (roughly 1850 - present i) that there is a new regime that starts at around 20- 30 years: the true climate regime. At longer times, the fluctuations increase rather than decrease, so that this scale marks 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 retrospect, it is hard to escape the conclusion that the choice of thirty year climate normals was simply fortuitous. Indeed, using preindustrial “multiproxies” described below, we will see that in the 1930’s; the destabilizing anthropogenic warming was still too small to detect. For example, In section 6.? (fig. ?), we find that over the entire climate normal (1900-1930), only an increase of about 0.1oC global anthropogenic warming had occurredj - barely above the year to year natural variability (about 0.2 oC); even since 1750 the total warming was only about 0.3 oC. To the IMO scientists, the constancy of the first climate normal period would have been quite plausible.

A straightforward estimate of the macroweather-climate transition scale is given by the time that it takes for the anthropogenic warming to equal the typical natural variability, in the 1930’s, this was about 60 years. Since then, emissions and other anthropogenic forcings have greatly increased, so that today the time it takes for the anthropogenic warming to exceed the natural variability has been reduced to only about 16 – 18 years. As a consequence, the 30 year weather-macroweather transition time (fig?, ?) is in fact no more than an average over the recent epoch (since roughly 1880).

i 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.j From 1750; the current warming is about 1 oC.

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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 k (top) as functions of latitudel. The thick curves show the mean over all the longitudes, and the dashed lines are the longitude to longitude variationsm. 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 thirty year period, we can use it to define 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

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

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differences (anomalies) 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 intermittency of the climate states. Although there are not enough climate states (four or five at 28 year scales), to analyse their 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) associated with coastlines and mountain ranges.

Fig. 5.6: The absolute east-west 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 20C 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 vertical scales.

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

let’s take a closer look at macroweather. Since we defined macroweather as the regime where H<0, we must find out how it varies from place to place? Fig. 5.7 shows its spatial distribution. As we move from one region to another, its value is not at all constant but its range is limited to the interval -1/2 to 0. This rqange is theoretciqally very significant, first consider the lower limit, H = -1/2. This corresponds to taking a sequence of independent random numbersn - a “white noise” - the larger H’s up to H = 0 are obtained by increasingly smoothing them. Whereas white noise obviously cannot be predicted any better than a coin tossing – i.e. not at all - in the chapter 6? we will see that the more it is smoothed, the more neighbouring values are likely to be the same, the more predictable the process becomes. The upper limit of the empirical range, H = 0 limito is even infinitely predictablep. It is therefore significant that in the figure, we can see that the oceans tend to have a typical

n 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 chosen should be identical and they should each be chosen independently.o Of course an infinite amount of past data would be needed to fully exploit it!

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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 landq.

In order to get an idea of what typical series with different H values 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, there is a sequence of independent values with no relation between them (H = -1/2, white noise)r. As one moves from bottom to top, H increases. Although fluctuations still tend to cancel, due to correlations between successive values, the cancelation is less and less perfect so that for the H’s that are near zero (top); there are coherent undulations present over most of the record.

We can now graphically illustrate a basic consequence of H<0; the sensitivity of the results to the degree of averaging; the temporal resolution. The original 1680 point long series (corresponding to 140 years) is shown in black; superposed is the same series but averaged over a factor of 12 in scale (blue; at annual resolution) and finally, the averaged and rescaled series is shown 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 12H in order to compensate for this averaging, to statistically correct for it. As expected, the averaging without compensation (blue) can be very strongs, the compensation does good job at restoring the range of variability so that it is comparable to the variability at the original one month resolution.

Although this strong resolution dependence is a basic feature of macroweather, it is neither recognized nor explicitly taken into account in either GCM model validation, nor in empirical temperature estimates. In the case of monthly and longer GCM series, there is a glaring symptom: the amplitudes of month to month GCM temperature variations are typically smaller than those in the data. But instead of recognizing this as a resolution issuet due to the nominal GCM resolution being lower (more averaged) than the data at the same nominal resolution; it is simply treated as yet another model imperfection to be corrected by ad hoc “post-processing”, now going by the sophisticated term “quantile matching”u. Unsurprisingly, the same resolution issue also 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. 5oX5o), the raw station and ship data are placed on the grid, and all the

p Predictability is the limit to which a process can be predicted using the best possible method with the best possible information; as we explain in see ch. ? there are two different predictability limits that are important: the more familiar deterministic predictability limit (the “butterfly effect”) and the (longer) stochastic predictability limits that are relevant here.q It turns out that spatial averages over large regions leads to subregions with H closest to zero dominating the others so that global averages – dominated by oceans - have H ≈ -0.1, see fig. ?r This statement is exactly true for the simulation, for the data it is only approximately true, being based on an estimate of H.s For the series at the bottom with H = -1/2, the amplitude is reduced by the factor 121/2 = 3.46. The factors (12-

H), are: 1.28, 1.64, 2.11, 2.70, 3.46 top to bottom.t The resolution may have a “mismatch” in both space and in time. The exact spatial resolution of GCMs is somewhat problematic because of the use of artificial “hyperviscous” damping/smoothing needed to keep the model numerically stable. u Quantile matching is a general method of forcing the probability distribution of the data and simulations to match. For Gaussian processes (a good approximation here), it is simply multiplication of 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).

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data in the grid square are averagedv 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 whatsoeverw! While over half a dozen such instrumentally based, centennial length data sets have been developed, each handles the gridding and missing data issues differently and this results in different effective resolutions, none of which are exactly the same as the nominal resolution of their gridded productsx (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 detailsy.

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

Center for Environmental Prediction (NCEP) reanalysis; monthly anomaly data from 1948-2015. Reproduced from Del Rio Amador et al 2017.

v The exact way that this is done varies depending on the approach, the data set; it never corresponds to a true areal average.w The fraction is somewhat variable, depending on the data set in question, but 50% is typical.x The average magnitude of the effect is about 11%, see box 5.1.y Global temperature estimates are based on these gridded products so that any multiplicative bias in the gridded data is propagated to the global averages.

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Fig. 5.8: A comparison of fractional Gaussian noise simulations (left) and selected 20C reanalysis anomalies (right) with the fluctuation exponent H increasing from - 1/2 to - 1/10 in steps of 1/10 (bottom to top)z. 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; corresponding to averaging from monthly to annual resolutions) and then – to statistically compensate for the cancellation – they are rescaled by multiplying by 12-H (red).

Fig. 5.9a: Temperature anomalies (left to right) for the 1000 month period from 1926 (left) to 2010 (right) as a function of longitude (every 6o along the 45oN meridian, from longitudes 0o to 354o

bottom to top), from the 20C reanalysis, each series was offset in the vertical. Notice that the fluctuation amplitudes as well as exponents H depend on position.

z In addition to removing the standard thirty year average annual cycle, we have also removed a linear trend and hence much of the global warming signal.

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Fig. 5.9b: A stochastic macroweather simulation using complex cascades displaying fluctuations with amplitudes and H’s that depend on position (bottom to top) both are random.

The geographical distribution of H values (fig. 5.7) is only part of the spatial variability; when we move from one region to another, not only does the H change, but the amplitude of the fluctuations also changes. Fig. 5.9a shows a typical space-time plot; even a visual impression of the density of black shows that the amplitudes are far from uniform from one location to another.

Figures 5.7, 5.8 underline the fact that 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 little is yet known, there are theoretical reasons (from cascade models), numerical reasons (from the analysis of GCM outputs) and empirical reasons (from temperature and precipitation data), to suppose that macroweather obeys a symmetry known 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), explaining the spatial variability of the amplitudes in fig. 5.9a. However, the only way to combine factorization with variable exponents H turns out to be to make space-time models in which the H values themselves are random, fig. 5.9b shows one such method based on cascadesaa.

The factorization principle is already implicitly used by weather services to “homogenize” data setsbb or to produce various climate indices that allow regions with different climates to be compared. Take the example of Portugal with a north ten times wetter than the south. We would like to compare monthly precipitation series to determine if they are both undergoing a similar long term climate changes. In the dry south, a change of 10 mm/month of rainfall over a ten year period would be highly significant, whereas in the wetter region, would only correspond to expected natural

aa Technically, the figure shows a “complex cascade”, one involving the multiplication of complex numbers; a variant of models presented in:5 Schertzer, D. & Lovejoy, S. in Space/time Variability and Interdependance for various hydrological processes (ed R.A. Feddes) 153-173 (Cambridge University Press, 1995).bb For example, data from individual climate stations are made commensurate (“homogenized”) by using the station standard deviations or probability distributions to adjust them.

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variability. A consequence of factorization is that if we divide (normalize) each series cc by the typical amplitude of its fluctuations, that the two can then be compared on an equal footing (‘hence the term “homogenization”).

Another important application of factorization discussed in ch. ? is that, the temperatures, precipitation (or other variables) 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 by “teleconnections”, yet the Equator temperature would not necessarily improve forecasts of Montreal temperatures. This counterintuitive result allows 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, it is explained in ch. 6?.

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, Van der Hoven’s discovery of the “synoptic maximum” – spectacularly graphically misrepresented as a scalebound spectral bumpdd (section 4.?) -and its subsequent relegation to a minor scalebound role, was a huge lost opportunity for clarifying the meaning of “weather” and for giving it a satisfactory objective definition.

We have argued that to tame atmospheric variability, we need a scaling framework and figs. ?, ? showed that out to ice age scales that 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 lower frequency regimes? One option would be to insist that the climate immediately follows the weather. In this case, slow variations over scales of thirty years and longer must be something else: “macro” (“big”) climate? This term would lead to some bizarre usages. For example, the “climate” would change from month to month and current monthly predictions would be “climate forecasts”ee. When referring to periods thirty or more years in the past; we would talk about “past macroclimates”, ice cores and other paleo proxy data would be “macroclimate indicators”, and the fight against global warming would be a struggle to stop “macroclimate change”.

To be consistent with common parlance, we should probably reserve the term “climate” for the third, longer time scale 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 governing the atmosphere, similarly, cascade models are based on the

cc These statements are true of the precipitation anomalies, after the annual cycle and long term average has been removed. dd 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. ee Admittedly, the current term “long range weather forecasts” is untenable and needs to be replaced by “macroweather forecasts”, ch. 7?.

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corresponding emergent higher level statistical laws. Bothff were originally weather models that were later extended to include the oceans, the cryosphere, and the carbon cycle and both can be used to objectively define the climates of models.

Let us focus on the more highly developed of the two 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 its 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 errors7 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 can no longer be interpreted deterministically because any small (microscopic) error would double and rapidly destroy the forecastgg.

Since it is a model, GCMs can be integrated forward from their initial conditions with fixed solar output, fixed atmospheric composition (e.g. Greenhouse gases), no volcanism and no land use changes (e.g. deforestation): constant external conditions. Runs with everything external to the atmosphere fixed are called “control runs”. In control runs, the atmosphere only displays “internal variability”, it varies in a quasi-steady manner around a long-term state: the model “climate”. In control runs directly observes the model’s convergence to its 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, the figure thus displays the model convergence to its climate, the rate being quantified by its slope, H. Since H ≈ -0.15hh, this convergence is “ultra slow”8 and from fig. 5.9, we can see that after 300 simulated years, the global temperature still typically fluctuates by 0.1oC aboutii its long term climatejj. This value of H near zero implies that if the simulations were extended to one million years, that averages would still typically fluctuate by 0.03 oC from its true 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 control runs thus accurately reproduce the basic weather and macroweather statistics.

ff 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:6 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). gg 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. hhThis is the mean in fig. 5.9 but there is some variation from model to model.ii This means that rerunning the same GCM for 300 simulated years only with tiny differences in the starting conditions, would lead to a 300 average about 0.1 oC from the first.jj The ultra slow convergence leads to technical difficulties in determining 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 that are attributed to deep (and slow) ocean currents (they cannot be attributed to global warming since all the external conditions are fixed). In fig. 5.9, drifts were approximately removed by linear regression: only the residuals after removing a straight line were analyzed.

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Similarly, turbulent cascade models that were designed to reproduce the weather statistics (including multifractal intermittency), also reproduce the weather and macroweather, although with a somewhat more negativekk H (fig. 5.9). When extended to scales beyond the lifetimes of planetary structures these (essentially) weather models thus naturally produce macroweather so that we can finally conclude that the “macroweather” not “microclimate” is the indeed appropriate term.

Fig. 5.9: Top: (red), the Haar fluctuations for 11 control runs from the Climate Model Intercomparison Project 5 (CMIP5)ll. The reference slope H = -0.15.Bottom: Haar fluctuations from cascade simulations, at resolution of 1 day for 340 years. The reference slope is H= -0.32 (multifractal simulations).

5.3 Modelling the climate: the climate of models So what about climate change? From the modellers point of view, the only way to

change the (model) climate is by changing the solar output, by allowing volcanoes to erupt and by accounting for changing levels of greenhouse gases and other human interventions: by changing the boundary conditions. In the words of Bryson9 “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” mm. 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.nn

kk More sophisticated models can reproduce different H values, see appendix 10D in:2 Lovejoy, S. & Schertzer, D. The Weather and Climate: Emergent Laws and Multifractal Cascades. (Cambridge University Press, 2013).ll Selected for length and absence of over active El Nino (3- 5 year variability).mm Pielke has criticized 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:13 Pielke, R. Climate prediction as an initial value problem. Bull. of the Amer. Meteor. Soc. 79, 2743-2746 (1998).nn This including the recent idea of “seamless forecasting” in which seasonal GCM forecasts are used to try to improve longer term climate predictions e.g.: 10 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); 11 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). 12 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).

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Since changing the boundaries force the model to (eventually) evolve to a new climate, the changed conditions “force” the climate. Factors responsible for this forcing are thus called “climate forcings”. This can sometimes be a bit confusing since for example, the standard solar radiation reference level is not a “climate forcing”. However, the 1000 times smaller deviations from this level due to sunspots or other solar activity are climate forcings.

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 defining the climate in terms of models is highly academic. When it comes to the real world, there are problems. An obvious one is that GCM modellers confidently assume that all processes relevant to determining the climate state are in the model, that none have been forgotten. What if – due to our limited understanding of the climate system, the model missed out on some slow decadal, centennial or millennial scale process?

But even if all the “internal;” processes are included and realistically modelled, there is another problem: none of the natural climate forcings are ever constant for long enough for the ultra slow model convergence to take place. Taking the important examples of volcanic and solar variability, these are themselves highly variable over decadal, centennial and millennial scales (they are also scaling!), and it is the way that their variability increase or decreases as a function of time scale that is important (the signs of their own H values!). Due to constant perturbations, the control run defined climate state might never even be approached so that its relevance would be questionable. For example, if these forcings decreased in amplitude at longer and longer times, they might not prevent convergence to a climate state (i.e. H might still be <0), but they could still yield a model climate different from the real one (in this case, we would need the equivalent of a control run but with volcanic activity reaching into the distant past to determine the long term effect on the climate state). However, if the missing forcings were strong enough, - especially if they increased in amplitude at long times (H>0)- they could provide destabilizing internal sources of variability and cause the climate state to slowly vary, to “wander”oo. In this case no amount of long term simulations would result in any convergence to any climate defined in this way. In conclusion, the current control run based concept of climate states may not even be relevant to GCMs when they are forced by unstable forcings with H>0, certainly they are not very helpful for defining the real world climate. In contrast, our scaling based definition of climate - as the H>0 regime beyond macroweather - is empirically accessible, it requires no models and yet it can be applied to the models when they are given appropriate climate forcings.

In view of the impossibility of actually implementing a control run definition of climate, GCM validation is difficult. One of the approaches is to make long historical based simulations since the year 1000 AD: “Last Millenium” simulations. Fig. 5.10a, b shows the result when this is done using NASA’s E2-R model. To make the simulations, one needs to use appropriate boundary conditions, notably, solar output, volcanic eruptions, and more recently, land use changes, greenhouse gases, and aerosols (pollution)pp. For most of the period since 1000, estimates of these “forcings”qq are in themselves highly challenging:

oo Of course, ultimately at very, very long scales even these forgotten internal processes might themselves exhibit a converging new macroclimate regime.pp By astronomical standards, 1000 years is a short time so that the orbital parameters of the earth have changed little, they are taken as fixed.qq The term “forcing” is a little confusing because the when the boundary conditions are fixed in control runs, the atmosphere is of course forced by the sun, however the term “forcing” used in this context refers to perturbations from the control run conditions that –given long enough time – would change the climate state.

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direct instrumental records don’t exist, the forcings must be indirectly infered from paleo indicators; we discuss this in section?

Given the forcings, the model still needs to be “spun up”. This is the technical term for the fact that the model initial state can only be guessed at; the model itself is run for a long period (here 150 simulated years, from 850 AD) in order that the model generates plausible atmospheric and more importantly, oceanic flows; this long period simply in order to start the trustworthy part of the simulation is needed because of the existence of deep ocean currents that take a very long time to develop realistic flow patterns; it is a symptom of the ultra slow model convergence. Nvertheless, just as with the control runs, even after the spin up period, the model tends to “drift” notably with temperatures spuriously changing slowly over long periods, in Last Millenium simulations, this is approximately corrected by ad hoc “post processing”.

The NASA E2-R Last Millenium runs were divided into industrial and preindustrial periods using 1880 as the dividing line8. Although this is somewhat arbitrary, it corresponds to the beginning of the period of steep increase in fossil fuel use and also to the period where the instrumental record begins to be more reliable and complete. Fig. 5.10a shows the E2-R Haar fluctuations over the industrial period (thin lines) compared with those from an instrumental data set, global and northern hemisphere (thick lines). The three simulations all include the same Greenhouse gases and solar reconstructions, they are distinguished by three different natural forcing scenarios, the two different volcanic reconstructions (top: “Gao” and “Crowley” for their authors, described below), and the bottom with only solar variability, no volcanoesrr.

By comparing the results with the instrumental fluctuations, (solid), we see that the variability of the simulations is fairly realistic and that it doesn’t make much difference which volcanic reconstructions are used – or, the bottom thin line - whether any are used at all. From the figure we conclude that since 1880, the Greenhouse gases dominate the natural forcings, and that the time scale at which the natural fluctuations become dominated by these anthropogenic forcings – the industrial epoch macroweather to climate transition scale - is pretty realistic. Also shown for reference is an estimate of the preindustrial variability deduced from temperature proxies (discussed later), that shows that the industrial epoch variability is indeed quite different from the pre-industrial one.

Now compare this with the same Last Millenium simulations but over the pre-industrial epoch, notably with virtually no Greenhouse gas forcings (fig. 5.10b; i.e. the atmospheric composition is fixed to pre-industrial levels). We see that at scales up to about a century, that the volcanic induced variability is much too high, and then at scales of several centuries that strong volcanic forcings are not enough; the long time variability is too small. We also see that by itself, the solar forcing is so small that it is essentially indistinguishable from the control run which by definition has no forcings at all. If the thick black line marked “multi-proxies 1500-1900” representing ground truth is to be believed, the pre-industrial macroweather-climate transition occurs at about centennial scales (the vertical dashed lines in fig. 5.10a), so that there is a problem of inadequate multicentennial variability: discussed below.

rr Actually, for each of the three shown, there were sub-variants with different land use scenarios, but these made very little difference.

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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 E2-R 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. Adapted from 8.

Fig. 5.10b: Haar fluctuation analysis of globally, annually averaged outputs of past Millenium simulations over the pre-industrial period (1500-1900) using the NASA GISS E2-R 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, the reference line has slope -0.2, nearly the same as that in fig. 5.7 which is the average over many different model control runs. Adapted from 8.

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5.4. Did civilization arise thanks to freak Holocene macroweather? “The long, stable Holocene is a unique feature of climate during the past 420 kyr,

with possibly profound implications for evolution and the development of civilizations.” So concluded an influential analysis14 of the 3300 m long Vostok (Antarctic) ice core record that spans 420,000 years, four complete ice age cycles (see fig. 1.?), directly linking stable temperature variations since the end of the last ice age (the Holocene) with the development of civilization. Presumably, this link was made because farming developed less than a thousand years after the retreat of the ice sheets and agriculture was crucial for transforming society.

The stability conclusion was based on eyeballing the Vostok record over the recent period. Its uniqueness was noted by comparing it with similar periods earlier in the same record. Determining the Haar fluctuations from the same data, fig. 5.11 (green) confirms the stability - we see that the unstable H<0 regime extends at least to several millennia implying a preindustrial macroweather – climate transition at multimillennial scalesss. For comparison, the figure also shows the Haar fluctuations averaged over the previous 410,000 years of the same core, showing (unstable) H>0 behaviour for scales as small as 100 yearstt. Turning to another long ice core – this time from Greenland (the GRIP core at 5 year resolutionuu), it turns out that the Holocene is apparently even more exceptional (fig. 5.11, blue), with the pre-Holocene GRIP and Vostok curves agreeing remarkably well with each other. The radical Holocene uniqueness can be confirmed simply by visual inspection: fig. 5.12 shows four earlier 10,000 year segments (top, black) each with strong variability, each “wandering” (due to H>0) over a total range of about 5oC, quite different from the fifth, (Holocene) series (brown).

ss Other shorter Greenland cores confirm that the GRIP results are typical of the Greenland Holocene and scaling spectral analyses agree with the Haar analysis presented here, see e.g.:15 Blender, R., Fraedrich, K. & Hunt, B. Millennial climate variability: GCM simulation and Greenland ice‐ cores. Geophys. Res. Lett., 33, , L04710, doi:doi:10.1029/2005GL024919 (2006).tt As we go back into the past we are using information from deeper and deeper parts of the core and these increasingly suffer from compression and ultimately molecular diffusion that spoils the distinct layers that define the temporal resolution. A general feature of the Antarctic is that it is extremely dry so that snow accumulates very slowly; in general the resolution of temperature proxies is not very high, here around 100 years. Dust does not suffer from the same problem and has thus been measured at decadal time scales back 800000 years in another Antarctic core (EPICA).uu The same one shown in fig. 1.?.

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Fig. 5.11: A comparison the RMS Haar fluctuations for both Vostok and GRIP cores at resolutions of 5 and 50 years respectively over the last 90 kyrs. These series were broken into 10 kyr sections. The dashed lines show the most recent of these (roughly the Holocene), the [Berner et al., 2008] paleo SST series is the upper dashed black line. The solid blue and green lines are of the ensemble of the eight 10 – 90 kyr GRIP and Vostok cores. Adapted from [Lovejoy and Schertzer, 2012].

Fig. 5.12: The top part shows four successive 10 kyr sections of the 5 yr resolution GRIP data, the most recent to the oldest from (red) bottom to top. Each series is separated by 10 mils in the vertical for clarity (vertical units: mils - i.e. parts per thousand of isotope excess), for reference, a 5 K corresponding temperature spread is also shown using a calibration constant of 0.5 K/mil. We see that the bottom Holocene GRIP series is indeed relatively devoid of low frequency variability compared to the previous 10 kyr sections, (fig. 5.11). In contrast, the bottom curve shows (the much lower resolution but on the same scale) paleo SST curve from ocean core LO09-14 [Berner et al., 2008], taken from a location only 1500 km distant and displaying far larger variability. Adapted from [Lovejoy and Schertzer, 2012].

Yet, before concluding that the Holocene is miraculously stable, consider a final core, this time from ocean sediments off the coast off Greenland and only 1500km from Summit Greenland where the GRIP core was taken (bottom, blue fig. 5.12). The ocean sediment temperature proxy is based on the 18O isotopes in layers of planktonic “foraminifera” that sank to the ocean bottom after living their lives in near surface waters taking their temperature signal with them. Although the temporal resolution of the data is lower (about 100 years), its character, including its range of temperature variations (fig. 5.11, 5.12) is very similar to the pre-Holocene ice cores. The difference between this Sea Surface Temperature (SST) proxy and the ice proxies – its patently unstable character - led the authors to conclude that the Holocene was on the contrary “highly unstable”16.

Both ice core and ocean proxies are generally regarded as robust, trustworthy paleo indicators, so which is right? Have our species been spoiled by a long, blissful macroweather hiatus, or – on the contrary - did harshly varying climate adversity force us to invent new ways of coping?

Keeping to the climate side of the debate, there may be a simple resolution of this apparent divergence. What if the ice cap Holocene climate is simply not representative of global conditions? What if the transition from macroweather to the climate varies strongly

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from place to place? In order to answer this, we need global scale, not regional scale temperature proxies. Although the exact time scale of the transition may be poorly discerned, we may still be certain that macroweather does indeed eventually give way to the climate. We know that at scales of 50,000 years (half a glacial-interglacial cycle), that the temperature varies by ±2 to ±3 oC (i.e. a total range of 4 - 6 oC, somewhat less near the equator) so that the typical Holocene fluctuations in fig. 5.11 must increase rapidly at times scales only a little larger than those shown; the glacial-interglacial window of fig. ? and fig. 5.? below.

In 2015, the question of Holocene stability and its region variations were a main theme at a workshop that I organized with Anne de Vernal in Jouvence Quebec. The interest that it sparked in understanding and quantifying centennial and millennial Holocene variability led to its support by the international past climate organization PAGES. A series of three PAGES workshops is currently attempting to bring together experts on paleo data with nonlinear geoscience specialists in order to better answer this question.

***

The key development that transformed climate science - and that can potentially answer the question about the Holocene stability - was the development in 1998 by Mann, Bradley and Hughes of “multiproxy reconstructions”17,18. Up until then, numerous individual climate series from tree rings (dendrochronology), varves (lake sediments), palynology (pollen, dust), ice cores, Mg/Ca variation in shells, 18O in foraminifera, diatoms, speleothems (stalagmites), biota and other sources had began to proliferate leading to numerous local reconstructions of past climates. Unsurprisingly, they were all highly variable in time and in space and the chronologies and calibrations is terms of temperatures were poorly discerned. The wide sweep of global climate change was only qualitative and quite Eurocentric; largely based on historical evidencevv such as records from the middle ages of vines growing in Britain, or Breugel’s 16th century renditions of skaters on frozen Dutch canals in the, respectively illustrating the “medieval warming event” and the “little ice age”.

Up until the advent of multiproxies, while each individual proxy series may have had some climate informationww they were noisy and the calibration required knowledge of the temperatures at the proxy sites over decades and longer and these were rarely available. In addition, although neighbouring proxies could potentially back each other up to reject noise and to better detect systematic regional changes, there was no way of statistically combining them. The multiproxy breakthrough was the use of a technique called “Empirical Orthogonal Functions” (EOFs) that allowed the instrumental record to determine the main components (EOFs) of the temperature variability so as to both reject noise and by determining the temperature at locations where there were no direct measurements to fill in data “holes”.

By using hundreds of proxy series, including (annual resolution) dendrochronologies, Mann et al thus developed the first quantitative reconstruction of northern hemisphere temperatures, first since 1500 AD17, but soon extended back to the year 1000 AD18. The result was the instantly acclaimed “hockeystick”, a graphic showing a gradual decline of northern hemisphere temperatures since 1000 AD, followed by a (relatively) rapid

vv At the time, the classic reference was Horace Lamb’s monumental “Past Present and Future”:19 Lamb, H. H. Climate: Past, Present, and Future. Vol. 1, Fundamentals and Climate Now . 613 pp. (Methuen and Co., 1972).:ww For simplicity, we restrict our attention to temperature proxies, but others, notably precipitation, do exist.

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warming since the end of the 1800’s (see fig. 1.? for a version of this); the period of long decline was the stick handle, the recent warming was the business end. The hockeystick xx was the first quantification of the industrial epoch warming and it visually underlined its rapidity. This lead to the famous conclusion – showcased in the 2001 IPPC AR3 21 - that the 20thC was the warmest century of the millennium that the 1990’s was the warmest decade and that 1998 was the warmest year. It foisted its feisty lead author, Michael Mann into centre stage of the developing “Climate Wars” that we will discuss in more detail in the next chapter.

Consideration of the original series 17 (extended back to 1000 AD in 18) illustrates both the technique and its attendant problems. The problem was the longer time scales, the low frequencies. A basic difficulty was in getting long series that are both temporally uniform and spatially representative. For example, the original six-century long multiproxy series presented in Mann et al 1998 had 112 indicators going back to 1820, 74 to 1700, 57 to 1600 and only 22 to 1400. Since only a small number of the series go back more than two or three centuries, the series’ “multicentennial” variability depends critically on how one takes into account the loss of data at longer and longer time intervals. A somewhat different low frequency issue is that the EOF technique that is critical for calibrating the proxies and rejecting noise only works properly if H<0, yet the global warming meant that H>0 at scales 20 - 30 years and longeryy. This means that today’s EOFs are effectively a little bit different from those of the pre-industrial past and this can potentially lead to errors in extrapolating current calibrations into the past.

Once the basic technique for producing multiproxies was known, they rapidly proliferated, so that by 2002, there were already five covering the northern hemisphere. Unsurprisingly, the technique was intensely scrutinized by climate skeptics (notably 22, 23) who spent considerable effort trying to destroy the hockeystick. Pressure from critics, meant that increasing attention was paid to the treatment of the low frequencies. One way to do this is to use borehole data which – when combined with the use of the equation of heat diffusion has essentially no calibration issues whatsoever. 24 used 696 boreholes (only back to 1500 AD, beyond this, the resolution becomes poor)zz. Similarly, in order to give proper weight to proxies with decadal and lower resolutions, (especially lake and ocean sediments), Moburg et al25 used wavelets to separately calibrate the low and high frequency proxies. Once again the result was a series with increased low frequency variability. Finally, Ljunquist26 used a more up to date and more diverse collection of proxies to produce a decadal resolution series going back to 1 AD.

We can visually compare several of the reconstructions; fig. 5.13; showsthat more recent (those published after 2003, thick) generally have considerably larger overall temperature variations than the earlier (pre 2003) multiproxies (top). Quantifying this using Haar fluctuations; we see in fig. 5.14 that up to scales of 150 years or so, that they all agree pretty well with each other, but for scales longer than a century or two, that the more recent multiproxies are more variable at long times. There are two arguments in their favour: first, one of them is the borehole based series discussed above and since it is a direct physical measurement, this avoids all the usual calibration issues. The second is that fig. 5.14 also shows that the low frequency variability seems to be more or less what is

xx At first, this was a climate skeptic term of derision, but it was soon adopted by Mann himself; see his excellent:20 Mann, M. E. The Hockey Stick and the Climate Wars: Dispatches from the Front Lines . 448 (Columbia University Press, 2012).yy This point is still not clearly appreciated.zz In order to obtain annual resolution, these authors did use dendrochronology, but the decadal and longer scales were from the Boreholes.

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needed in order to explain the lower frequency Vostok paleodata - assuming of course that the global scale (northern hemisphere) Holocene has the same type of variability as the previous epochs. These multiproxies are consistent with the hypothesis that at global scales, the Holocene is statistically much like the previous periods. Also shown in the figure is the industrial epoch instrumental based Haar fluctuations confirming that they are too strong to be compatible with the preindustrial multiproxies, industrial epoch warming is indeed unprecedented (see ch. ?).

Fig. 5.13: Nine multiproxy reconstructions published since 1998, from 1500- 1900 and smoothed to 30 years resolution to bring out the low frequencies. The multiproxies are divided into two groups; at the top the pre 2003 and the bottom (shifted by 0.3 oC for clarity) are four post 2003 multiproxies. The names correspond to the first authors on the papers where they were described. All the series have been shifted so as to agree on the temperature in the year 1900 AD. At the left, we can clearly see the wide range of the inferred temperature change since 1500 AD and the fact that generally, the post 2003 multiproxies are more widely varying.

Fig. 5.14:

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References:

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 Schertzer, D. & Lovejoy, S. in Space/time Variability and Interdependance for various hydrological processes (ed R.A. Feddes) 153-173 (Cambridge University Press, 1995).

6 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).

7 Lorenz, E. N. The predictability of a flow which possesses many scales of motion. Tellus 21, 289–307 (1969).8 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).

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

10 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).

11 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).

12 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).

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

14 Petit, J. R. et al. Climate and Atmospheric History of the Past 420,000 years from the Vostok Ice Core, Antarctica. Nature 399, 429-436 (1999).

15 Blender, R., Fraedrich, K. & Hunt, B. Millennial climate variability: GCM‐simulation and Greenland ice cores. Geophys. Res. Lett., 33, , L04710, doi:doi:10.1029/2005GL024919 (2006).

16 Berner, K. S., N., K., Divine, D., Godtliebsen, F. & Moros, M. A decadal-scale Holocene sea surface temperature record from the subpolar North Atlantic

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constructed using diatoms and statistics and its relation to other climate parameters Paleoceanography 23, doi:Doi:10.1029/2006pa001339 (2008).

17 Mann, M. E., Bradley, R. S. & Hughes, M. K. Global-scale temperature patterns and climate forcing over the past six centuries. Nature 392, 779-787 (1998).

18 Mann, M. E., Bradley, R. S. & Hughes, M. Northern Hemisphere Temperatures During the past Millenium: Inferences, Uncertainties, and Limitations. Geophys. Res. Lett. 26, 759-762 (1999).

19 Lamb, H. H. Climate: Past, Present, and Future. Vol. 1, Fundamentals and Climate Now. 613 pp. (Methuen and Co., 1972).

20 Mann, M. E. The Hockey Stick and the Climate Wars: Dispatches from the Front Lines. 448 (Columbia University Press, 2012).

21 Houghton, J. T. et al. (Cambridge University Press, 2001).22 McIntyre, S. & McKitrick, R. Corrections to the Mann et al (1998) proxy data

base and northern hemispheric average temperature series. Energy Environ. 14, 751-771 (2003).

23 McIntyre, S. & McKitrick, R. Hockey Sticks, Principal components and spurious signficance. Geophys. Resear. Lett. 32, L03710-L03714 (2005).

24 Huang, S. Merging Information from Different Resources for New Insights into Climate Change in the Past and Future. Geophys.Res, Lett. 31, L13205, doi:doi : 10.1029/2004 GL019781 (2004).

25 Moberg, A., Sonnechkin, D. M., Holmgren, K., Datsenko, N. M. & Karlén, W. Highly variable Northern Hemisphere temperatures reconstructed from low- and high - resolution proxy data. Nature 433, 613-617 (2005).

26 Ljundqvist, F. C. A new reconstruction of temperature variability in the extra - tropical Northern Hemisphere during the last two millennia. Geografiska Annaler: Physical Geography 92 A, 339 - 351, doi: DOI : 10.1111/j .1468 - 0459.2010 .00399.x (2010).

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