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Low frequency weather and the emergence of the Climate 1

S. Lovejoy1, D. Schertzer2,3 2

1 Physics, McGill University, 3600 University St., Montreal, Que. H3A 2T8, Canada 3 2 LEESU, Ecole des Ponts ParisTech, Université Paris Est, France 4

3 Météo France, 1 Quai Branly, Paris 75005, France 5 6

Abstract 7

We survey atmospheric variability from weather scales up to several 100 kyrs. We focus 8 on scales longer than the critical τw ≈ 5- 20 day scale corresponding to a drastic transition from 9 spectra with high to low spectral exponents. Using anisotropic, intermittent extensions of 10 classical turbulence theory, we argue that τw is the lifetime of planetary sized structures. At τw 11 there is a dimensional transition at longer times, the spatial degrees of freedom are rapidly 12 quenched, leading to a scaling “low frequency weather” regime extending out to τc ≈ 10 -100 yrs. 13 The statistical behaviour of both the weather and low frequency weather regime are well 14 reproduced by turbulence - based stochastic models and by control runs of traditional GCM’s, 15 i.e. without the introduction of new internal mechanisms or new external forcings, hence it is still 16 fundamentally “weather”. Whereas the usual (high frequency) weather has a fluctuation 17 exponent H > 0 implying that fluctuations increase with scale, in contrast, a key characteristic of 18 low frequency weather is that H < 0 so that fluctuations decrease instead. Therefore, it appears 19 “stable” and averages over this regime (i.e. up to τc) define climate states. However, at scales 20 beyond τc - whatever the exact causes - we find a new scaling regime with H>0 i.e. where 21 fluctuations again increase with scale, climate states thus appear unstable, this regime is thus 22 associated with our notion of climate change. 23

We use spectral and difference and Haar structure function analyses of reanalyses, 24 multiproxies and paleotemperatures. 25

26

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26

1. Introduction 27

1.1. What is the climate? 28

Notwithstanding the explosive growth of climate science over the last twenty years, there 29 is still no clear universally accepted definition of what the climate is – or what is almost the same 30 thing – what the difference is between the weather and the climate. The core idea shared by 31 most climate definitions is famously encapsulated in the dictum: “The climate is what you 32 expect, the weather is what you get” (see (Lorenz, 1995) for a discussion). In more scientific 33 language “Climate in a narrow sense is usually defined as the "average weather," or more 34 rigorously, as the statistical description in terms of the mean and variability of relevant 35 quantities over a period of time ranging from months to thousands or millions of years 36 (Intergovernmental Panel on Climate Change. Appendix I: Glossary. Retrieved on 2007-06-01). 37

An immediate problem with these definitions is that they fundamentally depend on 38 subjectively defined averaging scales. While the World Meteorological Organization defines 39 climate as 30 year or longer variability, a period of two weeks to a month is commonly used to 40 distinguish weather from climate so that even with these essentially arbitrary periods, there is 41 still a range of about a factor 1000 in scale (30 years /2 weeks) that is up in the air. This fuzzy 42 distinction is also reflected in numerical climate modelling since Global Climate Models are 43 fundamentally the same as weather models but at lower resolutions, with a different assortment 44 of subgrid parametrisations and they are coupled to ocean models - and increasingly – to 45 cryosphere, carbon cycle and land use models. Consequently, whether we define the climate as 46 the long - term weather statistics, or in terms of the long- term interactions of components of the 47 “climate system”, we still need an objective way to distinguish it from the weather. These 48 problems are compounded when we attempt to objectively define climate change. 49

However, there is yet another problem with this and allied climate definitions: they imply 50 that climate dynamics are nothing new: that they are simply weather dynamics at long time 51 scales. This seems naïve since we know from physics that when processes repeat over wide 52 enough ranges of space or time scale, qualitatively new climate laws should emerge from the 53 higher frequency weather laws. These “emergent” laws could simply be the consequences of 54 long range statistical correlations in the weather physics in conjunction with qualitatively new 55 climate processes – due to either internal dynamics or to external forcings - their nonlinear 56 synergy giving rise to emergent laws of climate dynamics. 57

Since the atmosphere is a nonlinear dynamical system with interactions and variability 58 occurring over huge ranges of space and time scales (millimetres to planet scales, milliseconds to 59 billions of years, ratios ≈1010 and ≈1020 respectively), the natural approach is to consider it as a 60 hierarchy of processes each with wide range scaling, i.e. each with nonlinear mechanisms that 61 repeat scale after scale over potentially wide ranges. Following (Lovejoy and Schertzer, 1986), 62 Schmitt et al., 1995, Pelletier, 1998, (Koscielny-Bunde et al., 1998), (Talkner and Weber, 63 2000), (Blender and Fraedrich, 2003), (Ashkenazy et al., 2003; Huybers and Curry, 2006b; 64 Rybski et al., 2008) this approach is increasingly superseding earlier approaches that postulated 65 more or less white noise backgrounds with a large number of spectral “spikes” corresponding to 66 many different quasi-periodic processes. This includes the slightly more sophisticated variants 67

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e.g. (Mitchell, 1976) which retain the spikes but replace the white noise with a hierarchy of 68 Ornstein-Uhlenbeck processes (white noises and their integrals); in the spectrum, “spikes” and 69 “shelves”; see also (Fraedrich et al., 2009) for a hybrid which includes a single (short) scaling 70 regime. 71

72 Fig. 1a: A modern composite based only on two sources: the Summit GRIP core 73

(Greenland paleotemperatures) and the Twentieth Century (20CR) reanalyses at the same 74 latitude (75oN). All spectra have been averaged over logarithmically spaced bins, 10 per order of 75 magnitude and the 20CR spectra have been averaged over all 180 longitude, 2ox2o elements, 76 frequency units: (yrs)-1. The light green is the mean of the GRIP 5.2 resolution data for last 90 77 kyrs and the (lowest) frequency blue is from the lower (55 cm) resolution GRIP core interpolated 78 to 200 yr resolution and going back 240 kyrs. The solid reference lines have absolute slopes βlw 79 = 0.2, and βc = 1.4 and βw = 2 as indicated. The red arrows at the bottom (and upper right) 80 indicate the basic qualitatively different scaling regimes. Reproduced from (Lovejoy and 81 Schertzer, 2011). 82

83 Over the past 25 years, scaling approaches have also been frequently applied to the 84

atmosphere, mostly at smaller scales but in the last five years increasingly to global scales. This 85 has given rise to a new scaling synthesis covering the entire gamut of meteorological scales from 86 milliseconds to beyond the ≈10 day period which is the typical lifetime of planetary structures 87 i.e. the weather regime. In the review (Lovejoy and Schertzer, 2010), it was concluded that the 88 theory and data were consistent with wide range but anisotropic spatial scaling, and that the 89 lifetime of planetary sized structures, provides the natural scale at which to distinguish weather 90 and a qualitatively different lower frequency regime. Fig. 1a shows a recent composite indicating 91 the three basic regimes covering the range of time scales from ≈ 100 kyrs down to weather 92 scales. The label “weather” for the high frequency regime seems clearly justified, and requires 93 no further comment. Similarly the lowest frequencies correspond to our usual ideas of 94

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multidecadal, multicentennial, multimillennial variability as “climate”. However labelling the 95 intermediate region “ low frequency weather” - rather than say “high frequency climate” – needs 96 some justification. The point is perhaps made more clearly with the help of fig. 1b which shows 97 a blowup of fig. 1a with both global and locally averaged instrumentally based spectra as well 98 the spectrum of the output of the stochastic Fractionally Integrated Flux (FIF) model (Schertzer 99 and Lovejoy, 1987) as well as the spectrum of the output of a standard GCM “control run” i.e. 100 without special anthropogenic, solar, orbital or other climate forcings. This regime is therefore 101 no more than “low frequency weather”, it contains no new internal dynamical elements, nor any 102 new forcing mechanism. As we discuss below, whereas the spectra from data (especially when 103 globally averaged) begin to rise for frequencies below ≈ (10 yrs)-1, both the FIF and GCM 104 control runs maintain their gently sloping “plateau” – like behaviours out to at least (500 yrs)-1 105 (note that we shall see that the “plateau” is not perfectly flat but its logarithmic slope is small, 106 typically in the range -0.2 to -0.6). Similar conclusions for the control runs of other GCMs at 107 even lower frequencies were found by (Blender et al., 2006) and (Rybski et al., 2008) so that it 108 seems that in the absence of external climate forcings, the GCM’s reproduce the low frequency 109 weather regime but not the lower frequency spectrally rising regime which requires some new 110 climate ingredient. The aim of this paper is to understand the natural variability so that the 111 important question of whether or not GCM’s with realistic forcings might be able to reproduce 112 the low frequency climate regime is outside our present scope. Certainly existing studies of the 113 scaling of forced GCM’s (Vyushin et al., 2004), (Blender et al., 2006) and (Rybski et al., 114 2008), have consistently reported unique low frequency weather exponents and this, even at the 115 lowest simulated frequencies. 116

117

118 119

Fig. 1b: A comparison of the spectra of temperature fluctuations from the the GRIP Greenland 120 Summit core, last 90 kyrs, 5.2 yr resolution, (green, upper left), monthly 20CR reanalysis (blue), 121 global (bottom) and 2o resolution (top, at 75oN) and a 500 yr control run of the monthly Institut 122 Pierre Simon Laplace, (IPSL) GCM (brown) used in the 4th IPCC report, at the corresponding 123 resolutions (frequency units: (yrs)-1). The dashed lines are the detrended daily, grid scale data 124

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(75oN 20CR dashed blue) and the cascade based Fractionally Integrated Flux simulation (dashed 125 red) both adjusted vertically to coincide with the analyses of the other, monthly scale data. 126 Reference lines with slopes βc = 1.4, βlw =0.2, βw = 1.8 are shown. Notice that the IPSL control 127 run – which lacks external climate forcing and is therefore simply low frequency weather as well 128 as the low frequency extension of the cascade based FIF model – continue to have shallow 129 spectral slopes out to their low frequency limits whereas the globally averaged 20CR and paleo 130 spectra follow βc ≈ 1.4 to roughly their low frequency limits. Hence the plateau is best 131 considered “low frequency weather”: the true climate regime has a much steeper spectrum 132 determined either by new low frequency internal interactions or by the low frequency climate 133 “forcing” (solar, orbital, volcanic, anthropogenic or other), fig. 1a shows that the βc ≈ 1.4 regime 134 continues to ≈ (100 kys)-1. 135 136

2. Temporal scaling, weather, low frequency weather, and the 137 climate 138

2.1 Discussion 139

Although spatial scaling is fundamental for weather processes, time scales much greater 140 than τw ≈ 10 the spatial degrees of freedom essentially collapse (via a “dimensional transition”), 141 so that we focus on temporal variability (section 2.5). In order to simplify things as much as 142 possible in section 2, we will only use spectra. 143

Consider a random field f(t) where t is time. Its “spectral density” E(ω) is the average 144 total contribution to the variance of the process due to structures with frequency between ω and 145 ω + dω (i.e. due to structures of duration τ = 2π/ω where τ is the corresponding time scale). E(ω) 146 is thus defined as: 147

E !( ) = f !( )!2

(1) 148

where where f !( )! is the Fourier transform of f (t) and the angular brackets “< . >” indicate 149

statistical averaging. f t( )

2

is thus the total variance (assumed to be independent of time), so 150

that the spectral density thus satisfies f t( )

2= E(!)d!

0

"

#. 151

In a scaling regime, we have power law spectra: 152 E !( ) " !#$

(2) 153

If we now consider the real space reduction in scale by factor λ we obtain: !" #$1! 154

corresponding to a “blow up” in freqeuncies: !" #! ; a power law E(ω) (eq. 2) maintains its 155 form under this transformation: E! "#$

E so that E is “scaling” and the “spectral exponent” β is 156 “scale invariant”. If empirically we find E of the form eq. 2, we take this as evidence for the 157

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scaling of the field f. Note that numerical spectra have well known finite size effects; the low 158 frequency effects have been dealt with below using standard “windowing” techniques (here a 159 Hann window was used to reduce spectral leakage). 160

161

2.2 Temporal spectral scaling in the weather regime 162

One of the earliest atmospheric spectral analyses was that of (Van der Hoven, 1957) 163 whose graph is at the origin of the legendary “meso-scale gap”, the supposedly energy-poor 164 spectral region between roughly 10-20 minutes and ≈ 4 days (ignoring the diurnal spike). Even 165 until fairly recently, textbooks regularly reproduced the spectrum (often redrawing it on different 166 axes or introducing other adaptations), citing it as convincing empirical justification for the neat 167 separation between low frequency isotropic 2D turbulence - identified with the weather - and 168 high frequency isotropic 3-D “turbulence”. This picture was seductive since if the gap had been 169 real, the turbulence would be no more than an annoying source of perturbation to the (2-D) 170 weather processes. 171

However, it was quickly and strongly criticized (e.g. (Goldman, 1968), (Pinus, 1968), 172 (Vinnichenko, 1969), (Vinnichenko and Dutton, 1969 ), (Robinson, 1971) and indirectly by 173 (Hwang, 1970)). For instance, on the basis of much more extensive measurements (Vinnichenko, 174 1969) commented that even if the mesoscale gap really existed, it could only be for less than 5% of 175 the time; he then went on to note that Van der Hoven’s spectrum was actually the superposition of 176 four spectra and that the extreme set of high frequency measurements were taken during a single 177 one hour long period during an episode of ‘near-hurricane’ conditions and these were entirely 178 responsible for the high frequency “bump”. 179

More modern temporal spectra are compatible with scaling from dissipation scales to ≈ 5- 180 20 days. Numerous wind and temperature spectra now exist from milliseconds to hours and days 181 showing for example that β ≈ 1.6, 1.8 for v, T respectively, some of this evidence is reviewed in 182 (Lovejoy and Schertzer, 2010) (Lovejoy and Schertzer, 2011). Fig. 2 shows an example, the 183 hourly temperature spectrum for frequencies down to (4 yrs)-1. According to fig. 2, it is plausible 184 that the scaling in the wind holds from small scales out to scales of ≈ 5 -10 days where we see a 185 transition. This transition is essentially the same as the low frequency “bump” observed by Van 186 der Hoven; its appearance only differs because he used a ωE(ω) rather than logE(ω) plot. 187

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188 Fig. 2: This shows the scaling of hourly surface temperatures from 4 stations in the northwest 189 US, for 4 years (2005-2008) taken from the US Climate Reference Network. One can see that in 190 spite of the strong diurnal cycle (and harmonics), the basic scaling extends to about 7 days. The 191 reference lines (with absolute slopes 0.2, 2: these values are theoretically motivated, for low 192 frequency weather and weather scales respectively). The spectra of hourly surface temperature 193 data from 4 nearly colinear stations running north west - south east in the US (Lander, WY, 194 Harrison NE, Whitman NE, Lincoln NE), from the US Climate Reference Network, 2005-2008. 195 The yellow line is the raw spectrum, the thick line is the spectrum of the periodically detrended 196 spectrum, averaged over logarithmically spaced bins, 10 per order of magnitude. 197

2.3 Temporal spectral scaling in the low frequency weather- climate 198

regime 199

Except for the annual cycle, the roughly flat low frequency spectra of fig. 2 (and fig. 1 200 (between ≈ 10 days and 10 yrs) are qualitatively reproduced in all the standard meteorological fields 201 and the transition scale τw is relatively constant. Fig. 3 shows estimates of τw, estimated using 202 reanalyses taken from the Twentieth Century Reanalysis project, 20CR, (Compo et al., 2011), on 203 2ox2o grid boxes. Also shown are estimates of τc, the scale where the latter ends and the climate 204 regime begins. The factor ≈ 1000 between τw and τc is the low frequency weather regime. Also 205 shown are estimates of the planetary scale eddy turn over time discussed below. 206

Fig. 4 shows the surface air temperature analysis out to lower frequencies and compares this 207 with the corresponding spectrum for sea surface temperatures (section 2.7). We see that the ocean 208 behaviour is qualitatively similar except that the transition time scale τo is ≈ 1 yr and the “low 209 frequency ocean” exponent βlo ≈ 0.6 which is a bit larger than the corresponding βlw ≈ 0.2 for the 210 air temperatures ( “lw”, “lo” for “low” frequency “weather” and “ocean” respectively). For 211 comparison we also show the best fitting Orenstein-Uhlenbeck processes (essentially integrals of 212

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white noises, i.e. with βw = 2, βlw = 0), these are the basis of Stochastic Linear Modeling approaches 213 (e.g. (Penland, 1996)). 214

215 Fig. 3: The variation of τw (bottom curves) and τc (top curves) as a function of latitude as 216 estimated from the 138 year long 20CR reanalyses, 700 mb temperature field compared with 217 with the theoretically predicted planetary scale eddy turn over time (τeddy, thick blue) and the 218 effective external scale (τeff) of the temperature cascade estimated from the ECMWF interim 219 reanalysis for 2006 (thin blue). The τw estimates were made by performing bi-linear regressions 220 on spectra from 180 day long segments averaged over 280 segments per grid point. Red shows 221 the mean over all the longitudes, the dashed lines are the corresponding longitude to longitude 222 one standard deviation spreads. The τc were estimated by bi-linear fits on the Haar structure 223 functions applied to the same data but monthly averaged. 224

225 To underline ubiquity of the low frequency weather regime, its low β character and to 226

distinguish it from the higher frequency weather regime, this regime was called the “spectral 227 plateau” (Lovejoy and Schertzer, 1986), although it is somewhat of a misnomer since it is clear that 228 the regime has a small but nonzero logarithmic slope whose negative value we indicate by βlw. The 229 transition scale τw was also identified as a weather scale by (Koscielny-Bunde et al., 1998). 230 231

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232 Fig. 4: This figure superposes the ocean and atmospheric plateaus showing their great similarity. 233 Left: A comparison of the monthly SST spectrum (bottom, blue) and monthly atmospheric 234 temperatures over land (top, purple) for monthly temperature series from 1911-2010 on a 5ox5o 235 grid; the NOAA NCDC data). Only those near complete series (missing less than 20 months out 236 of 1200) were considered; 465 for the SST, 319 for the land series; the missing data were filled 237 using interpolation. The reference slopes correspond to β = 0.2 (top), 0.6 bottom left and 1.8, 238 bottom right. A transition at 1 year corresponds to a mean ocean εο ≈ 1x10-8m2s-3. The dashed 239

orange lines are Orenstein-Uhlenbeck processes (of the form E !( ) = "2/ !

2+ a

2( ) , σ, a are 240 constants) used as the basis for Stochastic Linear Forcing models. 241 Right: The average of 5 spectra from 6 year long sections of a thirty year series of daily 242 temperatures at a station in France (black, taken from (Lovejoy and Schertzer, 1986)). The red 243 reference line has a slope 1.8. The relative up-down placement of this daily spectrum with the 244 monthly spectra (corresponding to a constant factor) was determined by aligning the atmospheric 245 spectral plateaus (i.e. the black and purple spectra). The raw spectra are shown (no averaging 246 over logarithmically spaced bins). 247

2.4 The weather regime, space-time scaling and some turbulence theory 248 In order to understand the weather and low frequency weather scaling, we briefly recall 249

some turbulence theory using the example of the horizontal wind v. In stratified scaling 250 turbulence (the 23/9D model (Schertzer and Lovejoy, 1985a)), the energy flux ε dominates the 251 horizontal and the buoyancy variance flux φ dominates the vertical so that horizontal wind 252 fluctuations Δv (e.g. differences, see section 3) follow: 253

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!v !x( ) = "1/3!xHh ; Hh = 1 / 3 3a

!v !y( ) = "1/3!yHh ; Hh = 1 / 3 3b

!v !z( ) = #1/5!zHv ; Hv = 3 / 5 3c

!v !t( ) = "1/2!t H$ ; H$ = 1 / 2 3d

(3) 254

where Δx, Δy, Δz, Δt are the increments in horizontal coordinates, vertical coordinate and time 255 respectively and the exponents Hh, Hv, Hτ are “fluctuation” or “non conservation” exponents in 256 the horizontal, vertical and in time respectively. Since the mean fluxes are independent of scale 257 (i.e. <ε>, <φ> are constant, “<.>” indicates ensemble averaging) these exponents express how the 258 fluctuations Δv increase (H>0) or decrease (H<0) with the scale (i.e. the increments). Equations 259 (3 a - b) describe the real space horizontal (Kolmogorov, 1941) scaling and 3 c the vertical 260 Bolgiano-Obukhov (BO, (Bolgiano, 1959), (Obukhov, 1959)) scaling for the velocity. The 261 anisotropic Corrsin-Obukov law for passive scalar advection is obtained by the replacements 262 v! "; # ! $ 3/2#%1/2 where ρ is the passive scalar density, χ is the passive scalar variance flux 263 (Corrsin, 1951), (Obukhov, 1949). 264

Before proceeding, a few technical comments. In eq. 3, the equality signs should be 265 understood in the sense that each side of the equation has the same scaling properties: the 266 Fractionally Integrated Flux model is essentially a more precise interpretation of the equations in 267 terms of fractional integrals of order H. Ignoring intermittency (associated with multifractal 268 fluxes which we discuss only briefly below), the spectral exponents are related to H as β =1+2H 269 so that βh =5/3, βv =11/5, βτ=2. Finally, although the notation “H” is used in honour of E. Hurst, 270 in multifractal processes it is generally not identical to the Hurst (i.e. rescaled range, “R/S”) 271 exponent, the relationship between the two is nontrivial. 272

Although these equations originated in classical turbulence theory, the latter were all 273 spatially statistically isotropic so that the simultaneous combination of the horizontal laws 3a, b 274 with the vertical law 3c is nonclassical. Since the isotropy assumption is very demanding, the 275 pioneers originally believed that the classical laws would hold over only scales of scales of 276 hundreds of meters. The anisotropic extension implied by eqs. 3 a-d is itself based on a 277 generalization of the notion of scale invariance - thus has the effect of radically changing the 278 potential range of validity of the laws. For example, even the finite thickness of the troposphere 279 - which in isotropic turbulence would imply a scale break at around 10 km – no longer implies a 280 break in the scaling. Beginning with (Schertzer and Lovejoy, 1985b), it has been argued that 281 atmospheric variables including the wind do indeed have wide range (anisotropic) scaling 282 statistics (see the review (Lovejoy and Schertzer, 2010)). 283

In addition, the classical turbulence theories were for spatially uniform (“homogeneous”) 284 turbulence in which the fluxes were quasi constant (e.g. with Gaussian statistics). In order for 285 these laws to apply up to planetary scales, starting in the 1980’s, (Parisi and Frisch, 1985), 286 (Schertzer and Lovejoy, 1985b) they were generalized to strongly variable (intermittent) 287

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multiplicative cascade processes yielding multifractal fluxes so that although the mean flux 288 statistics <εΔx> remain independent of scale Δx, the statistical moments: 289

<εΔxq> ≈ Δx-K(q) (4) 290

where εΔx is the flux averaged over scales Δx and K(q) is the (convex) moment scaling exponent. 291 Although K(1) = 0, for q ≠1, K(1) ≠0 so that <εΔx

q> is strongly scale dependent, the fluxes are 292 thus the densities of singular multifractal measures. 293

Along with the spatial laws (eqs. 3 a- c), we have included eq. 3 d which is the result for 294 the pure time evolution in the absence of an overall advection velocity; this is the classical 295 Lagrangian version of the Kolmogorov law (Inoue, 1951; Landau and Lifschitz, 1959), it is 296 essentially the result of dimensional analysis using ε and Δt rather than ε and Δx. Although 297 Lagrangian statistics are notoriously difficult to obtain empirically (see however (Seuront et al., 298 1996)), they are roughly known from experience and are used as the basis for the space-time or 299 “Stommel” diagrams that adorn introductory meteorology textbooks (see (Schertzer et al., 1997), 300 (Lovejoy et al., 2000) for scaling adaptations). 301

Due to the fact that the wind is responsible for advection, the spatial scaling of the 302 horizontal wind leads to the temporal scaling of all the fields. Unfortunately, space-time scaling 303 is somewhat more complicated than pure spatial scaling. At meteorological time scales this is 304 because we must take into account the mean advection of structures and the Galilean invariance 305 of the dynamics. The effect of the Galilean invariance / advection of structures is that the 306 temporal exponents in the Eulerian (fixed earth) frame become are the same as in the horizontal 307 direction (i.e. in (x,y,t) space we have “trivial anisotropy”, i.e. with “effective” temporal 308 exponent Hτeff = Hh). At the longer time scales of low frequency weather, the scaling is broken 309 because the finite size of the earth implies a characteristic lifetime (“eddy-turn-over time”) of 310 planetary scale structures. Using eq. 3 a,b with Δx = Le we obtain: τeddy = Le/Δv(Le) = εw

-1/3Le2/3 311

where Le = 20,000 km is the size of the planet and εw is the globally averaged energy flux 312 density. 313

Considering time scales longer than τeddy, the effect of the finite planetary size implies 314 that the spatial degrees of freedom become ineffective - there is a “dimensional transition” – so 315 that instead of interactions in (x,y,z,t) space for long times, the interactions are effectively only in 316 t space and this implies a drastic change in the statistics, summarized in the next section. 317

2.5 Low frequency weather and a dimensional transition 318

To obtain theoretical predictions for the statistics of atmospheric variability at time scales 319 τ >τw (i.e. in the low frequency weather regime), we can take the FIF model that produces 320 multifractal fields respecting eqs. 3 and extend it to processes with outer timescales τc>>τw. The 321 theoretical details are given in (Lovejoy and Schertzer, 2010) and (Lovejoy and Schertzer, 322 2011) but the upshot of this is that we expect the energy flux density ε to factor into a 323 statistically independent space-time weather process εw(r,t) and a low frequency weather process 324 εlw(t) which is only dependent on time: 325

! r,t( ) = !wr,t( )!

lwt( ) (5) 326

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In this way the low frequency energy flux εlw(t) - which physically is the result of nonlinear 327 radiation/ cloud interactions – multiplicatively modulates the high frequency space-time weather 328 processes. 329

The statistical behaviour of εlw(t) is quite complex to analyze and has some surprising 330 properties. Some important characteristics are: a) at large temporal lags Δt the autocorrelations 331 <εlw(t) εlw(t-Δt) > ultimately decay as Δt-1, although very large ranges of scale may be necessary 332 to observe it, b) since the spectrum is the Fourier transform of the autocorrelation and the 333 transform of a pure Δt-1 function has a low (and high) frequency divergence, the actual spectrum 334 of the low frequency weather regime depends on its overall range of scales Λc= τc/τw: from fig. 3, 335 we find that to within a factor of ≈ 2, that the mean Λc over the latitudes ≈ 1100, c) over 336 surprisingly wide ranges (factors of 100 – 1000 in frequency for values of Λc in the range 210 - 337 216), one finds “pseudo-scaling” with nearly constant spectral exponents βlw which are typically 338 in the range 0.2 – 0.4, d) the statistics are independent of H and only weakly dependent on K(q). 339

In summary, we therefore find for the overall turbulence (FIF) model: 340

E k( ) ! k"#w ;

E $( ) ! $"#w ;

E $( ) ! $"#lw ;

k > Le

"1

$ > %w

"1

%c

"1< $ < %

w

"1

(6) 341

where k is the modulus of the horizontal wave vector, τc is the long external scale where the low 342 frequency weather regime ends (see discussion below) and βl, βlw are: 343

!w= 1+ 2H

w" K 2( )

0.2 < !lw< 0.6

(7) 344

In the low frequency weather regime, the intermittency (characterized by K(q)) decreases as we 345 average the process over scales >τw so that the low frequency weather regime has an effective 346 fluctuation exponent Hlw: 347

Hlw! " 1" #

lw( ) / 2 (8) 348

The high frequency weather spectral exponents βw, Hw are the usual ones, but the low frequency 349 weather exponents βlw, Hlw are new. Since βlw<1, we have Hlw<0 and using 0.2 < βlw < 0.6 350 corresponding to -0.4 < Hlw < -0.2, this result already explains the preponderance of spectral 351 plateau β’s around that value already noted. However, as we saw in fig. 4, the low frequency 352 ocean regime has a somewhat high value βoc ≈ 0.6 (Hoc ≈ -0.2); (Lovejoy and Schertzer, 2011) 353 use a simple coupled ocean - atmosphere model to show how this could arise as a consequence 354 of double (atmosphere and ocean) dimensional transitions. 355

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2.6 The transition time scale from weather to low frequency weather using 356

“first principles” 357

Figures 1 – 4 show evidence that temporal scaling holds from small scales to a transition 358 scale τw of around 5 -20 days. Let us now consider the physical origin of this scale. In the 359 famous (Van der Hoven, 1957) ωE(ω) versus logω plot, its origin was argued to be due to 360 “migratory pressure systems of synoptic weather- map scale”. The corresponding features at 361 around 4 – 20 days notably for temperature and pressure spectra were termed “synoptic maxima” 362 by (Kolesnikov and Monin, 1965), and (Panofsky, 1969) in reference to the similar idea that it 363 was associated with synoptic scale weather dynamics, see (Monin and Yaglom, 1975) for some 364 other early references. 365

More recently, (Vallis, 2010) suggested that τw is the basic lifetime of baroclinic 366

instabilities which he estimated using the inverse Eady growth rate: !Eady " Ld /U where the 367

deformation rate is: Ld = Ri / f0 , f0 is the Coriolis parameter and Ri is the Richardson number 368

across the troposphere (= N2/s2 where N is the mean Brunt-Vaisalla frequency and s the mean 369 shear). The Eady growth rate is obtained by linearizing the equations about a hypothetical state 370 with both uniform shear and stratification across the troposphere. By taking Ri ≈104, f0 ≈ 10-4 371 rad/s, and U ≈ 10 m/s, one obtain Vallis’s estimate Ld ≈ 1000 km. Using the maximum Eady 372 growth rate theoretically introduces a numerical factor 3.3 so that the actual predicted inverse 373 growth rate is: 3.3τEady ≈ 4 days. Vallis similarly argues that this also applies to the oceans but 374 with U ≈ 10 cm/s and Ld ≈ 100 km yielding 3.3τEady ≈ 40 days. The obvious theoretical problem 375 with using τEady to estimate τw, is that the former is expected to be valid in homogeneous, quasi-376 linear systems whereas the atmosphere is highly heterogeneous with vertical and horizontal 377 structures (including strongly nonlinear cascade structures) extending throughout the troposphere 378 to scales substantially larger than Ld. Another difficulty is that although the observed transition 379 scale τw is well behaved at the equator (fig. 3), f0 vanishes implying that Ld and τEady diverge: 380 using τEady as an estimate of τw is at best a midlatitude approximation. Finally, the choice of 381 values used to estimate τEady are not obvious. For example, the estimate Ri ≈ 104 across the 382 troposphere seems quite high, and there is no evidence for any special behaviour at length scales 383 near Ld ≈ 1000 km. 384

If there is (at least statistically) a well defined relation between spatial scales and 385 lifetimes (the “eddy-turn-over time”), then the lifetime of planetary scale structures τeddy is of 386 fundamental importance. The shorter period (τ< τeddy) statistics are dominated by structures 387 smaller than planetary size whereas for τ > τeddy, they are dominated by the statistics of many 388 lifetimes of planetary scale structures. It is therefore natural to take τeddy ≈ τw. 389

In order to estimate τw we therefore need an estimate of the globally averaged flux energy 390 density εw. We can estimate εw by using the fact that the mean solar flux absorbed by the earth is 391 ≈200 W/m2 (e.g. (Monin, 1972)). If we distribute this over the troposphere (thickness ≈ 104 m), 392 with mean air density ≈ 0.75 Kg/m3, and we assume a 2% conversion of energy into kinetic 393 energy ((Palmén, 1959), (Monin, 1972)), then we obtain a value εw ≈ 5x10-4m2/s3 which is 394 indeed typical of the values measured in small scale turbulence (Brunt, 1939), (Monin, 1972). 395

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Using the ECMWF interim reanalysis to obtain a modern estimate of εw (Lovejoy and 396 Schertzer, 2010) showed that although ε is larger in mid latitudes than at the equator and that at 397 300 mb it reaches a maximum, the global tropospheric average is ≈ 10-3 m2/s3. They also showed 398 that the latitudinally varying ε, explained to better than ±20% the latitudinal variation of the 399 hemispheric antipodes velocity differences (using Δv = ε1/3Le

1/3) and concluded that the solar 400 energy flux does a good job of explaining the horizontal wind fluctuations up to planetary scales. 401 In addition, we can now point to fig. 3 which shows that the latitudinally varying ECMWF εw(θ) 402 estimates do indeed lead to τeddy(θ) very close to the 20CR τw(θ) estimates for the 700 mb 403 temperature field. 404

2.7 Ocean “weather” and “low frequency ocean weather” 405

It is well known that for months and longer time scales that ocean variability is important 406 for atmospheric dynamics; before explicitly attempting to extend this model of weather 407 variability beyond τw, we must therefore consider the role of the ocean. The ocean and the 408 atmosphere have many similarities; from the preceding discussion, we may expect analogous 409 regimes of “ocean weather” to be followed by an ocean spectral plateau both of which will 410 influence the atmosphere. To make this more plausible, recall that both the atmosphere and 411 ocean are large Reynolds’ number turbulent systems and both are highly stratified - albeit due to 412 somewhat different mechanisms. In particular there is no question that at least over some range, 413 horizontal ocean current spectra are dominated by the ocean energy flux εo. It roughly follows 414 that E(ω) ≈ω-5/3 and presumably in the horizontal: E(k) ≈ k-5/3; (i.e. βo = 5/3, Ho =1/3, see e.g. 415 (Grant et al., 1962), (Nakajima and Hayakawa, 1982)). Although surprisingly few current 416 spectra have been published, the recent use of satellite altimeter data to estimate sea surface 417 height (a pressure proxy) has provided relevant empirical evidence that k-5/3 continues out to 418 scales of at least hundreds of kilometers, refuelling the debate about the spectral exponent and 419 the scaling of the current, see (Le Traon et al., 2008). 420

Although empirically the current spectra (or their proxies) at scales larger than several 421 hundred kilometres are not well known, other spectra - especially those of sea surface 422 temperatures (SST) - are known to be scaling over wide ranges and due to their strong nonlinear 423 coupling with the current they are relevant. Using mostly remotely sensed infrared radiances, 424 and starting in the early 1970’s, there is much evidence for SST scaling up to thousands of 425 kilometres with β ≈ 1.8 -2 i.e. nearly the same as for the atmospheric temperature (see e.g. 426 (McLeish, 1970), (Saunders, 1972), (Deschamps et al., 1981), (Deschamps et al., 1984), 427 (Burgert and Hsieh, 1989), (Seuront et al., 1996), (Lovejoy et al., 2000), and a review in 428 (Lovejoy and Schertzer, 2011)). 429

If, as in the atmosphere, the energy flux dominates the horizontal ocean dynamics then 430 we can use the same methodology as in the previous subsection - basic turbulence theory (the 431 Kolmogorov law) combined with the mean ocean energy flux εo - to predict ocean eddy turn over 432 time and hence the outer scale τo of the ocean regime. Thus, for ocean gyres and eddies of size l, 433 we expect there to be a characteristic eddy turnover time (lifetime) τ = ε-1/3l2/3 with a critical 434 “ocean-weather” – “ocean-climate” transition time scale obtained when l = Le: τo = εo

-1/3Le2/3. 435

Again, we expect a fundamental difference in the statistics for fluctuations of duration τ< τo - the 436 ocean equivalent of “weather” with a turbulent spectrum with roughly βo ≈ 5/3 (at least for the 437

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current) - and for durations τ > τo, the ocean “climate” with a shallow ocean spectral regime with 438 β ≈ <1. Since the spatial β for temperature in the atmosphere and ocean are very close, if the β‘s 439 for the current and wind are also close, then so will the β for the temporal temperature spectra. 440

In order to test this idea, we need the globally averaged ocean current energy flux, εo. As 441 expected, εo is highly intermittent (see (Robert, 1976), (Clayson and Kantha, 1999), (Moum 442 et al., 1995), (Lien and DʼAsaro, 2006), (Matsuno et al., 2006)) and as far as we know, the 443 only attempt to estimate its global average is (Lovejoy and Schertzer, 2011) who used ocean 444 drifter maps of eddy kinetic energy. They found that εo ≈10-8 m2/s3 is a reasonable global 445 estimate for the surface layer (it decreases quite rapidly with depth). Using the formula τo = 446 εo

-1/3Le2/3 and εο in the range 1x10-8 - 8x10-8 m2s-3 we find τo ≈ 1 - 2 yrs; c.f. the values for the 447

atmosphere (εw ≈10-3m2s-3, τw ≈ 10 days). 448 This provides us with a prediction for the SST spectrum: E(ω)≈ ω-1.8 for ω ≈≥(1 year)-1 449

followed by a transition to a much flatter plateau (here ≈ ω-0.6) for the lower frequencies, see fig. 450 6 which compares the ocean and air over land spectra. While the latter spectrum is – as expected 451 - essentially a pure spectral plateau (with βlw ≈ 0.2, the value cited earlier), we see that the SST 452 spectrum is essentially the same (βo ≈ βw ≈1.8) except that βlo ≈ 0.6 and τo ≈1 year. This basic 453 “cross-over” to an exponent βlo ≈ 0.6 was already noted by (Monetti et al., 2003) who estimated 454 it at 300 days. Note also the rough convergence of the spectra at about 100 yr scale which 455 implies that the land and ocean variability become equal and also the hint that there is a low 456 frequency rise in the land spectrum for periods> ≈ 30 yrs. 457

2.8 Other evidence for the spectral plateau 458

Various published scaling composites such as figs. 1a, b give estimates for the low 459 frequency weather exponent βlw, the climate exponent βc and the transition scale τc; they agree 460 on the basic picture while proposing somewhat different parameter values and transition scales 461 τc. For example (Huybers and Curry, 2006a) studied many paleoclimate series as well as the 462 60 year long NCEP reanalyses concluded that for periods of months up to about 50 years, the 463 spectra are scaling with midlatitude βlw’s larger than the tropical βlw; (their values are 0.37±0.05, 464 0.56±0.08. Many analyses in the spectral plateau regime have been carried out using in situ data 465 with the Detrended Fluctuation Analysis (DFA) method (Fraedrich and Blender, 2003; 466 Koscielny-Bunde et al., 1998), (Bunde et al., 2004), sea surface temperatures (Monetti et al., 467 2003) and ≈1000 year long Northern hemisphere reconstructions (Rybski et al., 2006) (see also 468 (Lennartz and Bunde, 2009) and (Lanfredi et al., 2009)); see (Eichner et al., 2003), for a 469 review of many scaling analyses and their implications for long term persistence / memory issue. 470 From the station analyses the basic conclusions of (Fraedrich and Blender, 2003) were that over 471 land, β ≈ 0-0.1 whereas over the ocean β ≈ 0.3; whereas (Eichner et al., 2003) found β ≈ 0.3 472 over land and using NCEP reanalyses, (Huybers and Curry, 2006a) found slightly higher 473 values and noted an additional latitudinal effect (β is higher at the equator). At longer scales, 474 (Blender et al., 2006) analysed the anomalous Holocene Greenland paleo temperatures finding 475 β ≈ 0.5, see section 4.2. Other pertinent analyses are of Global Climate Model outputs and 476 historical reconstructions of the Northern hemisphere temperatures which are discussed in detail 477 in section 4.3. Our basic empirical conclusions, in accord with a growing literature – particularly 478

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with respect to the temperature statistics are that β is mostly in range 0.2 – 0.4 over land and ≈0.6 479 over the ocean. 480

481

3. Climate change 482

3.1 What is climate change? 483

We briefly surveyed the weather scaling, focusing on the transition to the low frequency 484 weather regime for time scales longer than the lifetimes of planetary scale eddies at τw ≈ 5 -20 485 days. This picture was complicated somewhat by the qualitatively similar (and nonlinearly 486 coupled) transition from the analogous ocean “weather” to “low frequency ocean weather” at τo 487 ≈ 1 yr. Using purely spectral analyses, we found that these low frequency regimes continued 488 until scales of the order of τc ≈10 - 100 years, after which the spectra started to steeply rise, 489 marking the beginning of the true climate regime. While the high frequency regime clearly 490 corresponds to “weather”, we termed the intermediate regime “low frequency weather” since its 491 statistics are not only well reproduced with (unforced) “control” runs of GCM’s (fig. 1b) but also 492 by (stochastic, turbulent) cascade models of the weather when they are extended to low 493 frequencies. The term “climate regime” was thus reserved for the long times τ >τc

where the low 494 frequency weather regime gives way to a qualitatively different and much more variable regime. 495 The new climate regime is thus driven either by new (internal) low frequency nonlinear 496 interactions, or by appropriate low frequency solar, volcanic, anthropogenic - or eventually 497 orbital - forcing at scales τ >τc. 498

This three scale-range scaling picture of atmospheric variability leads to a clarification of 499 the rough idea that the climate is nothing more than long-term averages of the weather. It allows 500 us to precisely define a climate state as the average of the weather over the entire low frequency 501 weather regime up to τc (i.e. up to decadal or centennial scales). This paves the way for a 502 straightforward definition of climate change as the long term changes in this climate state i.e. of 503 the statistics of these climate states at scales τ>τc. 504

3.2 What is τc? 505

In figure 1, we gave some evidence that τc was in the range (10 yr)-1 to (100 yrs)-1, i.e. it 506 was near the extreme low frequency limit of instrumental data. We now attempt to answer this 507 with more certainty. Up until now, we primarily used spectral analysis since it is a classical, 508 straightforward technique whose limitations are well known and it was adequate for the purpose 509 of determining the basic scaling regimes in time and in space. We now focus on the low 510 frequencies corresponding to several years to ≈ 100 kyrs so that it is convenient to study 511 fluctuations in real rather than Fourier space. There are several reasons for this. The first is that 512 we are focusing on the lowest instrumental frequencies, and so spectral analysis provides only a 513 few useful data points - for example on data 150 years long, the time scales longer than 50 514 years are characterized only by three discrete frequencies ω =1, 2, 3: Fourier methods are 515

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“coarse” at low frequencies. The second is that in order to extend the analysis to lower 516 frequencies it is imperative to use proxies, and these need calibration: the mean absolute 517 amplitudes of fluctuations at a given scale enable us to perform a statistical calibration. The 518 third is that the absolute amplitudes are also important for gauging the physical interpretation and 519 hence significance of the fluctuations. 520

3.3 Fluctuations and structure functions 521

The simplest fluctuation is also the oldest, the difference: (Δv(Δt))diff = Δv(t+Δt)- Δv(t). 522 The corresponding statistical moments <Δvq> are the classical “generalized” structure functions. 523 According to eq. 3, the fluctuations follow: 524

!v = "!t!tH

(9) 525 where ϕΔt is a resolution Δt turbulent flux. From this we see that the statistical moments follow: 526

!v !t( )q

= "!tq !t qH # !t $ q( )

; $ q( ) = qH % K q( ) (10) 527

ξ(q) is the structure function with exponent and K(q) is the (multifractal, cascade) intermittency 528 exponent, eq. 4. The turbulent flux has the property that it is independent of scale Δt, i.e. the first 529 order moment < φΔt > is constant, hence K(1) = 0 and ξ(1) = H. The physical significance of H is 530 thus that it determines the rate at which fluctuations grow (H>0) or decrease (H<0) with scale Δt. 531 Since the spectrum is a second order moment, there is the following useful and simple relation 532 between real space and Fourier space exponents: 533

! = 1+ " 2( ) = 1+ 2H # K 2( ) (11) 534

The problem is that the mean difference cannot decrease with increasing Δt, hence 535 differences are clearly inappropriate when studying scaling processes with H<0: the differences 536 simply converge to a spurious constant depending on the highest frequencies available in the 537 sample. Similarly, when H>1, fluctuations defined as differences saturate at a large Δt 538 independent value; they depend on the lowest frequencies present in the sample. In both cases, 539 the exponent ξ(q) is no longer correctly estimated. The problem is that we need a definition of 540 fluctuations such that Δv(Δt) is dominated by frequencies ≈ Δt-1. 541

The need to more flexibly define fluctuations motivated the development of wavelets 542 (e.g. (Bacry et al., 1989), (Mallat and Hwang, 1992; Torrence and Compo, 1998)), and the 543 related Detrended Fluctuation Analysis technique (DFA, (Peng et al., 1994) and (Kantelhardt 544 et al., 2001), (Kantelhardt et al., 2002) for polynomial and multifractal extensions respectively). 545 In this context, the classical difference fluctuation is only a special case, the “poor man’s 546 wavelet”. In the weather regime, most geophysical H parameters are indeed in the range 0 to 1 547 (see e.g. the review (Lovejoy and Schertzer, 2010)) so that fluctuations tend to increase with 548 scale and this classical difference structure function is generally adequate. However, a prime 549 characteristic of the low frequency weather regime is precisely that H<0 (section 2.5) so that 550 fluctuations decrease rather than increase with scale, hence for studying this regime, difference 551 fluctuations are inappropriate. To change the range of H over which fluctuations are usefully 552 defined, one changes the shape of the defining wavelet, changing both its real and Fourier space 553

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localizations. In the usual wavelet framework, this is done by modifying the wavelet directly 554 e.g. by choosing the Mexican hat or higher order derivatives of the Gaussian etc., or by choosing 555 them to satisfy some special criterion. Following this, the fluctuations are calculated as 556 convolutions with Fast Fourier (or equivalent) numerical techniques. 557

A problem with this usual implementation of wavelets is that not only are the 558 convolutions numerically cumbersome, but the physical interpretation of the fluctuations is lost. 559 In contrast, when 0<H<1, the difference structure function gives direct information on the typical 560 difference (q =1) and typical variations around this difference (q = 2) and even typical skewness 561 (q =3) or typical Kurtosis (q =4) or - if the probability tail is algebraic – of the divergence of high 562 order moments of differences. Similarly, when -1<H<0 one can define the “tendency structure 563 function” (below) which directly quantifies the fluctuation’s deviation from zero and whose 564 exponent characterizes the rate at which the deviations decrease when we average to larger and 565 larger scales. These poor man’s and tendency fluctuations are also very easy to directly estimate 566 from series with uniformly spaced data and - with straightforward modifications - to irregularly 567 spaced data. 568

The study of real space fluctuation statistics in the low frequency weather regime 569 therefore requires a definition of fluctuations valid at least over the range -1<H<1. Before 570 discussing our choice - the Haar wavelet - let us recall the definitions of the difference and 571 tendency fluctuations; the corresponding structure functions are simply the qth moments. The 572 difference/ poor man’s fluctuation is thus: 573

!v !t( )( )diff

" #!tv ; #

!tv = v t + !t( ) $ v t( ) (12) 574

where δ is the difference operator. Similarly, the “tendency fluctuation” (Lovejoy and 575

Schertzer, 2011) can be defined using the series with overall mean removed: !v t( ) = v t( ) " v t( ) 576 with the help of the summation operator s by: 577

!v !t( )( )tend

=1

!t"!tsv ' ; sv ' = #v #t( )

#t $t

% (13) 578

!v !t( )( )tend has a straightforward interpretation in terms of the mean tendency of the data but is 579

useful only for -1<H<0. It is also easy to implement: simply remove the overall mean and then 580 take the mean over intervals Δt: this is equivalent to taking the mean of the differences of the 581 running sum. 582

We can now define the Haar fluctuation which is a special case of the Daubechies family 583 of orthogonal wavelets (see e.g. (Holschneider, 1995), for a recent application, (Ashok et al., 584 2010), and for a comparison with the related Detrended Fluctuation Analysis technique, see 585 (Koscielny-Bunde et al., 1998; Koscielny-Bunde et al., 2006)). This can be done by taking the 586 second differences of the mean: 587

!v !t( )( )Haar

=2

!t"!t /22s =

1

!ts t( ) + s t + !t( )( ) # 2s t + !t / 2( )( )

=2

!tv $t( )

t+!t /2% $t %t+!t

& # v $t( )t% $t %t+!t /2

&'

()

*

+,

(14) 588

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From this, we see that the Haar fluctuation at resolution Δt is simply the first difference of the 589 series degraded to resolution Δt/2. Although this is still a valid wavelet (although with the extra 590 normalisation factor Δt-1), it is almost trivial to calculate and (thanks to the summing) the 591 technique is useful for series with -1<H<1. 592

For pure scaling functions, the difference (1>H>0) or tendency (-1<H<0) structure 593 functions are adequate, and have obvious interpretations. The real advantage of the Haar 594 structure function is apparent for functions with two or more scaling regimes - one with H>0, 595 one with H<0. We shall see that ignoring intermittency, this criterion is the same as β<1 or β>1 596 hence (see e.g. fig. 1a) Haar fluctuations will be useful for the data analyzed which straddle - 597 either at high or low frequencies - the boundaries of the low frequency weather regime. 598

Is it possible to “calibrate” the Haar structure function so that the amplitude of typical 599 fluctuations can still be easily interpreted? To answer this, consider the definition of a “hybrid” 600 fluctuation as the maximum of the difference and tendency fluctuations: 601

!T( )hybrid

= max !T( )diff, !T( )

tend( ) (15) 602 the “hybrid structure function” is thus the maximum of the corresponding difference and 603 tendency structure functions and therefore has a straightforward interpretation. The hybrid 604 fluctuation is useful if a calibration constant C can be found such that: 605 606

!T !t( )hybrid

q" C

q!T !t( )

Haar

q

(16) 607 In a pure scaling process with -1<H<1, this is clearly possible since the difference or tendency 608 fluctuations yield the same scaling exponent. However, in a case with two or more scaling 609 regimes, this equality cannot be exact, but as we see this in the next section, it can still be quite 610 reasonable approximation. 611

3.5 Application of Haar fluctuations to global temperature series 612

Now that we have defined the Haar fluctuations and corresponding structure function, we 613 can use it to analyse a fundamental climatological series: the monthly resolution global mean 614 surface temperature. At this resolution, the high frequency weather variability is largely filtered 615 out and the statistics are dominated first by the low frequency weather regime (H<0), and then at 616 low enough frequencies by the climate regime (H>0). 617

Several such series have been constructed. The three we chose are the NOAA NCDC 618 (National Climatic Data Center) merged land air and sea surface temperature dataset 619 (abbreviated NOAA NCDC below, from 1880 on a 5ox5o grid, see (Smith et al., 2008) for 620 details), the NASA GISS (Goddard Institute for Space Studies) data set (from 1880 on a 2ox2o 621 (Hansen et al., 2010) and the HadCRUT3 data set (from 1850 to 2010 on a 5o x5o grid). 622 HadCRUT3 is a merged product created out of the HadSST2 (Rayner et al., 2006) Sea Surface 623 Temperature (SST) data set and its companion data set CRUTEM3 of atmospheric temperatures 624 over land. The NOAA NCDC and NASA GISS are both heavily based on the Global Historical 625 Climatology Network (Peterson and Vose, 1997), and have many similarities including the use 626 of sophisticated statistical methods to smooth and reduce noise. In contrast, the HadCRUTM3 627 data is less processed. Unsurprisingly, these series are quite similar although analysis of the scale 628 by scale differences between the spectra is interesting, see (Lovejoy and Schertzer, 2011). 629

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Each grid point in each data set suffered from missing data points so that here we 630 consider the globally averaged series obtained by averaging over all the available data for the 631 common 129 yrs period 1880 – 2008. Before analysis, each series was periodically detrended to 632 remove the annual cycle – if this is not done, then the scaling near Δt ≈ 1 yr will be artificially 633 degraded. The detrending was done by setting the amplitudes of the Fourier components 634 corresponding to annual periods to the “background” spectral values. 635

Fig. 5 shows the comparison of the difference, tendency, hybrid and Haar root mean 636 square (RMS) structure functions <ΔT(Δt)2>1/2; the latter increased by a factor C = 100.35 ≈ 2.2. 637 Before commenting on the physical implications, let us first make some technical remarks. It 638 can be seen that the “calibrated” Haar and hybrid structure functions are very close; the 639 deviations are ±14% over the entire range of nearly a factor 103 in Δt. This implies that the 640 indicated amplitude scale of the calibrated Haar structure function in degrees K is quite accurate, 641 and that to a good approximation, the Haar structure function can preserve the simple 642 interpretation of the difference and tendency structure functions: in regions where the 643 logarithmic slope is between -1 and 0, it approximates the tendency structure function whereas in 644 regions where the logarithmic slope is between 0 and 1, the calibrated Haar structure function 645 approximates the difference structure function. For example, from the graph we can see that 646 global scale temperature fluctuations decrease from ≈ 0.3 K at monthly scales, to ≈ 0.2 K at 10 647 yrs and then increase to ≈ 0.8 K at ≈ 100 yrs. All of the numbers have obvious implications 648 although note that they indicate the mean overall range of the fluctuations, so that for example 649 the 0.8 K corresponds to ±0.4 K etc. 650

From fig. 5 we also see that the global surface temperatures separate into two regimes at 651 about τc ≈ 10 yrs, with negative and positive logarithmic slopes = ξ(2)/ 2≈ -0.1, 0.4 for Δt<τc and 652 Δt> τc respectively. Since β = 1+ξ(2) (eq. 11) we have β ≈ 0.8, 1.8. We also analysed the first 653 order structure function whose exponent ξ(1) = H; at these scales the intermittency (K(2), eq. 4) 654 ≈ 0.03 so that ξ(2) ≈ 2H so that H ≈ -0.1, 0.4 confirming that fluctuations decrease with scale in 655 the low frequency weather regime but increase again at lower frequencies in the climate regime 656 (more precise intermittency analyses are given in (Lovejoy and Schertzer, 2011)). Note that 657 ignoring intermittency, the critical value of β discriminating between growing and decreasing 658 fluctuations (i.e. H<0, H>0) is β = 1. 659

660

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661 Fig. 5: A comparison of the different structure function analyses (root mean square, 662

RMS) applied to the ensemble of three monthly surface series discussed in ch. 10 (NASA GISS, 663 NOAA CDC, HADCRUT3), each globally and annually averaged, from 1881-2008 (1548 points 664 each). The usual (difference, poor man’s) structure function is shown (orange, lower left), the 665 tendency structure function (purple, lower right), the maximum of the two (“Hybrid”, thick, red), 666 and the Haar in blue (as indicated); it has been increased by a factor C =100.35 = 2.2, the resulting 667 RMS deviation with respect to the hybrid is ±14%. Reference slopes with exponents ξ(2)/2 ≈ 668 0.4, -0.1 are also shown (black corresponding to spectral exponents β = 1+ξ(2) = 1.8, 0.8 669 respectively). In terms of difference fluctuations, we can use the global root mean square 670 !T !t( )

21/2

annual structure functions (fitted for 129 yrs>Δt>10 yrs), obtaining 671 !T !t( )

21/2

≈0.08!t 0.33 for the ensemble, in comparison, (Lovejoy and Schertzer, 1986) found the 672

very similar !T !t( )21/2

≈ 0.077Δt0.4 using northern hemisphere data (these correspond to βc = 673 1.66, 1.8 respectively). 674

675 Before pursuing the Haar structure function let us briefly consider its sensitivity to 676

nonscaling perturbations; i.e. to nonscaling external trends superposed on the data which break 677 the overall scaling. Even when there is no particular reason to suspect such trends, the desire to 678 filter them out is commonly invoked to justify the use of special wavelets – or nearly 679 equivalently – of various orders of the MultiFractal Detrended Fluctuation Analysis technique 680 (MFDFA), (Kantelhardt et al., 2002). A simple way to produce a higher order Haar wavelet that 681 eliminates polynomials of order n is simply to iterate (n+1 times) the difference operator in eq. 682 14 for example, iterating three times yields the “quadratic Haar” fluctuation 683 !v !t( )( )

Haarquad=3

!ts t + !t( ) " 3s t + !t / 3( ) + 3s t " !t / 3( ) " s t " !t( )( )

. This fluctuation is sensitive to 684 structures of size Δt-1 – and hence useful - over the range -1<H<2 and it is blind to polynomials 685 of order 1 (lines). In comparison, the nth order DFA technique defines fluctuations using the 686 RMS deviations of the summed series s(t) from regressions of nth order polynomials so that 687

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quadratic Haar fluctuations are nearly equivalent to the quadratic MFDFA RMS deviations 688 (although these deviations are not strictly wavelets, note that the MFDFA uses a scaling function 689 ≈ Δv Δt hence with DFA exponent αDFA=1+H). Although at first sight the insensitivity of these 690 higher order wavelets to external trends may seem advantageous, it should be recalled that on 691 the one hand they only filter out polynomial trends - and not for example the more geophysically 692 relevant periodic trends - while on the other hand, even for this, they are “overkill” since the 693 trends they filter are filtered at all scales – not just the largest. Indeed, if one suspects the 694 presence of external polynomial trends, it suffices to eliminate them over the whole series (i.e. at 695 the largest scales), and then to analyse the resulting deviations using the Haar fluctuations. 696

Fig. 6 shows the usual (linear) Haar RMS structure function (eq. 14) compared to the 697 quadratic Haar and quadratic MFDFA structure functions. It can be seen that the latter two are 698 close to each other (after applying different calibration constants, see the figure caption), that the 699 low and high frequency exponents are roughy the same. However, the transition point has 700 shifted by nearly a factor of 3 so that overall they are rather different from the Haar structure 701 function and it is clearly not possible to simultaneously calibrate the high and low frequency 702 parts. The drawback with these higher order fluctuations is thus that we loose the simplicity of 703 interpretation of the Haar wavelet and – unless H>1 - we obtain no clear advantage. 704

705 Fig. 6: The same temperature data as fig. 5; a comparison of the RMS Haar structure function 706 (multiplied by 100.35 = 2.2), the RMS quadratic Haar (multiplied by 100.15 = 1.4) and the RMS 707 quadratic MFDFA (multiplied by 101.5 = 31.6). 708

4. The transition from low frequency weather to the climate 709

4.1 Intermediate scale multiproxy series 710

In section 2 we discussed atmospheric variability over the frustratingly short 711 instrumentally accessible range of time scales (roughly Δt< 150 yrs) and saw evidence that 712

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weakly variable low frequency weather gives way to a new highly variable climate regime at a 713 scale τc somewhere in the range 10- 30 years. In fig. 1 we already glimpsed the much longer 1 -714 100 kyr scales accessible primarily via ice core paleo temperatures (see also below); these 715 confirmed that - at least when averaged over the last 100 kyr or so - the climate does indeed have 716 a new scaling regime with fluctuations increasing rather than decreasing in amplitude with scale 717 (H>0). 718

Since the temporal resolution of the high resolution GRIP paleo temperatures was ≈ 5.2 719 yrs (and for the Vostok series, ≈100 yrs) these paleo temperature resolutions don’t greatly 720 overlap the instrumental range; it is thus useful to consider other intermediates: the “multiproxy” 721 series that have been developed following (Mann et al., 1998). Another reason to use 722 intermediate scale data is because we are living in a climate epoch which is exceptional in both 723 its long and short term aspects. For example consider the long stretch of relatively mild and 724 stable conditions since the retreat of the last ice sheets about 11.5 kyr ago, the “Holocene”. This 725 epoch is at least somewhat exceptional: it has even been suggested that such stability is a 726 precondition for the invention of farming and thus for civilisation itself (Petit et al., 1999). It is 727 therefore possible that the paleoclimate statistics averaged over series 100 kyrs or longer may not 728 be as pertinent as we would like for understanding the current epoch. Similarly, at the high 729 frequency end of the spectrum, there is the issue of “20th Century exceptionalism”, a 730 consequence of 20th century warming and the probability that at least some of it may be of 731 anthropogenic – not natural - origin. Since these affect a large part of the instrumental record, it 732 is problematic to use the latter as the basis for extrapolations to centennial and millennial scales. 733 In the following we try to assess both “exceptionalisms” in an attempt to understand the natural 734 variability in the last few centuries. 735

4.2 The Holocene exception: climate variability in time and in space 736

The high resolution GRIP core gives a striking example of the difference between the 737 Holocene and previous epochs in central Greenland (fig. 7). Even a cursory visual inspection of 738 the figure confirms the relative absence of low frequency variability in the current 10 kyr section 739 as compared to previous 10 kyr sections. To quantify this, we can turn to fig. 8 which compares 740 the RMS Haar structure functions for both GRIP (Arctic) and Vostok (Antarctic) cores for both 741 the Holocene 10 kyr section and, for the mean and spread of the eight earlier 10 kyr sections. 742 The GRIP Holocene curve is clearly exceptional, with the fluctuations decreasing with scale out 743 to τc ≈ 2 kyrs in scale and with ξ(2)/2 ≈ -0.3. This implies a spectral exponent near the low 744 frequency weather value β ≈ 0.4, although it seems that ξ(2)/2 ≈ 0.4 (β≈ 1.8) for larger Δt. The 745 main difference however is that τc is much larger than for the other series (see table 1 for 746 quantitative comparisons). The exceptionalism is quantified by noting that the corresponding 747 RMS fluctuation function (S(Δt)) is several standard deviations below the average of the 748 previous eight 10 kyr sections. In comparison (to the right in the figure) the Holocene period of 749 the Vostok core is also somewhat exceptional although less so: up to τc ≈ 1 kyr it has ξ(2)/2 ≈ -750 0.3 (β ≈ 0.4) and it is more or less within one standard deviation limits of its mean although τc is 751 still large. Beyond scales of ≈ 1 kyr its fluctuations start to increase; table 1 quantifies the 752 differences. We corroborated this conclusion by an analysis of the 2 kyr long (yearly resolution) 753 series from other (nearby) Greenland cores (as described in (Vinther et al., 2008)) where 754

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(Blender et al., 2006) also obtained β ≈ 0.2-0.4 and also obtained similar low β estimates for the 755 Greenland GRIP, GISP2 cores over the last 3 kyr. 756

While these analyses convincingly demonstrate that the Greenland Holocene was 757 exceptionally stable, nevertheless their significance for the overall natural variations of northern 758 hemisphere temperatures is not clear. For example, on the basis of paleo SST reconstructions 759 just southeast of Greenland (Andersen et al., 2004) (also (Berner et al., 2008)) it was 760 concluded that the latter region was on the contrary “highly unstable”. Using several ocean cores 761 as proxies, Holocene SST reconstructions were produced which included a difference between 762 maximum and minimum of roughly 6 K and “typical variations” of 1- 3 K. In comparison, from 763 fig. 8 we see that the mean temperature fluctuation deduced from the GRIP core in the last 10 764 kyrs is ≈ 0.2 K. However also from fig. 8 we see that the mean over the previous eight 10 kyr 765 sections was ≈ 1 - 2 K, i.e. quite close to these paleo SST variations (and about amount expected 766 in order to explain the glacial/ interglacial temperature swings, see section 5). These paleo SST 767 series thus underline the strong geographical climate variability effectively undermining the 768 larger significance of the Greenland Holocene experience. At the same time they lend support to 769 the application of standard statistical stationarity assumptions to the variability over longer 770 periods (e.g. to the relevance of spectra and structure functions averaged over the whole cores). 771

772 Fig. 7: Four successive 10 kyr sections of the high resolution GRIP data, the most recent to the 773 oldest from bottom to top. Each series is separated by 10 mils in the vertical for clarity (vertical 774 units: mils). The bottom Holocene series is indeed relatively devoid of low frequency variability 775 compared to the other 10 kyr sections, a fact confirmed by statistical analysis discussed in the 776 text and fig. 8. 777

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778 779 780

Fig. 8: This compares the RMS Haar structure function (S(Δt)) for both Vostok and 781 GRIP high resolution cores (resolutions 5.2 and 50 years respectively over the last 90 kyrs). The 782 Haar fluctuations were calibrated and are accurate to ±20%. For Vostok we used the Petit et al 783 calibration, for GRIP, 0.5 K/mil. The series were broken into 10 kyr sections. The orange show 784 the most recent of these (roughly the Holocene, top Vostok, bottom, GRIP) whereas the blue and 785 green are the mean of the eight 10 – 90 kyr GRIP and Vostok S(Δt). The one standard deviation 786 variations about the mean are indicated by dashed lines. Also shown are reference lines with 787 slopes ξ(2)/2 = -0.3, 0.2, 0.4 corresponding to β = 0.4, 1.4, 1.8 respectively. Although the 788 Holocene is exceptional for both series, for GRIP it is exceptional by many standard deviations. 789 For the Holocene we can se that τc ≈ 1 kyr for Vostok, and ≈ 2 kyr for GRIP, although for the 790 previous 80 kyrs, we see that τc ≈100 yrs for both. 791 792

H β Holocene 10-90 kyrs Holocene 10-90 kyrs

Range of regressions

100yr< Δt < 2 kyr

2 kyr< Δt < 10 kyr

100yr < Δt < 10

kyr

100yr< Δt < 10 kyr

ensemble

100 yr< Δt < 2 kyr

2 kyr< Δt < 10 kyr

100yr < Δt < 10

kyr

100yr< Δt < 10 kyr

ensemble GRIP -0.25 0.21 0.14±0.18 0.17 0.43 1.33 1.14±0.33 1.20 Vostok -0.40 0.38 0.19±0.28 0.31 0.18 1.76 1.29±0.51 1.49

793 Table 1: This compares various paleo exponents estimated using the Haar structure function over 794 successive 10 kyr periods. Vostok at 50 yr resolution, GRIP at 5.2 year resolution, all 795 regressions over the scale ranges indicated. The Holocene is the most recent period (0-10 kyr). 796 Note that while the Holocene exponents are estimates from individual series, the 10-90 kyr 797 exponents are the means of the estimates from each 10 kyr section and (to the right), the 798

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exponent of the ensemble mean of the latter. Note that the mean of the exponents is a bit below 799 the exponent of the mean indicating that a few highly variable 10 kyr sections can strongly affect 800 the ensemble averages. For the Holocene, the separate ranges < 2 kyr and Δt>2 kyr were chosen 801 because according to fig. 8, τc ≈1- 2 kyr. 802

4.3 Multiproxy temperature data, centennial scale variability and the 803

20th Century exception 804

The key to linking the long but geographically limited ice core series with the short but 805 global scale instrumental series are the intermediate category of “multiproxy temperature 806 reconstructions”. These series of northern hemisphere average temperatures pioneered by 807 (Mann et al., 1998), (Mann et al., 1999) have the potential of capturing “multicentennial” 808 variability over at least the (data rich) northern hemisphere. These series at typically annual 809 resolutions combine a variety of different data types ranging from tree rings, ice cores, lake 810 varves, boreholes, ice melt stratigraphy, pollen, Mg/Ca variation in shells, 18O in foraminifera, 811 diatoms, stalactites (in caves), biota and historical records. In what follows, we analyze eight of 812 the longest of these; see table 2 for some of statistical characteristics and descriptions. 813

Before reviewing the results, let us discuss some of the technical issues behind the 814 continued development of new series. Consideration of the original series (Mann et al., 1998) 815 (extended back to 1000 AD in (Mann et al., 1999)) illustrates both the technique and its 816 attendant problems. The basic difficulty is in getting long series that are both temporally 817 uniform and spatially representative. For example, the original six-century long multiproxy 818 series presented in Mann et al 1998 has 112 indicators going back to 1820, 74 to 1700, 57 to 819 1600 and only 22 to 1400. Since only a small number of the series go back more than two or 820 three centuries, the series’ “multicentennial” variability depends critically on how one takes into 821 account the loss of data at longer and longer time intervals. When it first appeared, the Mann et 822 al series created a sensation by depicting a “hockey-stick” shaped graph of temperature: with the 823 fairly flat “handle” continuing from 1000 AD until a rapid 20thC increase. This lead to the 824 famous conclusion - echoed in the 3rd IPPC report - that the 20thC was the warmest century of the 825 millennium, that the 1990’s was the warmest decade and that 1998 was the warmest year. This 826 success encouraged the development of new series using larger quantities of more geographically 827 representative proxies (Jones et al., 1998), by the introduction new types of data (Crowley and 828 Lowery, 2000), to the more intensive use of pure dendrochronology, (Briffa et al., 2001) or to 829 improved methodologies (Esper et al., 2002). 830

However, the interest generated by reconstructions also attracted criticism, in particular, 831 (McIntyre and McKitrick, 2003) pointed out several flaws in the (Mann et al., 1998) data 832 collection and in the application of the principal component analysis technique which been had 833 borrowed from econometrics. After correction, the same proxies yielded series with significantly 834 larger low frequency variability and included the reappearance of the famous “medieval 835 warming” period at around 1400 AD which had disappeared in the original. Later, an additional 836 technical critique (McIntyre and McKitrick, 2005) underlined the sensitivity of the methodology 837 to low frequency red noise variability present in the calibration data (the latter modelled this with 838 exponentially correlated processes probably underestimating that which would have been found 839 using long range correlated scaling processes). Other work in this period, notably by (von 840

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Storch et al., 2004) using “pseudo-proxies” (i.e. the simulation of the whole calibration process 841 with the help of GCM’s) similarly underlined the nontrivial issues involved in extrapolating 842 proxy calibrations into the past. 843

Beyond the potential social and political implications of the debate, the scientific upshot 844 was that increasing attention had to be paid to the preservation of the low frequencies. One way 845 to do this is to use borehole data which – when combined with the use of the equation of heat 846 diffusion has essentially no calibration issues whatsoever. (Huang, 2004) used 696 boreholes 847 (only back to 1500 AD, the limit of this approach) to augment the original (Mann et al., 1998) 848 proxies so as to obtain more realistic low frequency variability. Similarly, in order to give proper 849 weight to proxies with decadal and lower resolutions, (especially lake and ocean sediments), 850 (Moberg et al., 2005) used wavelets to separately calibrate the low and high frequency proxies. 851 Once again the result was a series with increased low frequency variability. Finally, (Ljundqvist, 852 2010) used a more up to date, more diverse collection of proxies to produce a decadal resolution 853 series going back to 1 AD. The low frequency variability of the new series was sufficiently large 854 that it even included a 3rd century “Roman warm period” as the warmest century on record and 855 permitted the conclusion that “the controversial question whether Medieval Warm Period peak 856 temperatures exceeded present temperatures remains unanswered” (Ljundqvist, 2010). 857

With this context, let us quantitatively analyse the eight series cited above; we use the 858 Haar structure function. We concentrate here on the period 1500-1979 because: a) it is common 859 to all eight reconstructions, b) being relatively recent, it is more reliable (it has lower 860 uncertainties) and c) it avoids the medieval warm period and thus the possibility that the low 861 frequency variability is artificially augmented by the possibly unusual warming in the 1400’s. 862 The result is shown in fig. 9 where we have averaged the structure functions into the five pre-863 2003 and three post-2003 reconstructions. Up to Δt ≈ 200 years the basic shapes of the curves 864 are quite similar to each other – and indeed to the surface temperature S(Δt) curves back to 1881 865 (fig. 1). However quite noticeable for the pre 2003 constructions is the systematic drop in RMS 866 fluctuations for Δt ≈>200 yrs which contrasts with their continued rise in the post 2003 867 reconstructions. This confirms the above analysis to the effect that the post 2003 analyses were 868 more careful in their treatments of multicentennial variability. Table 2 gives a quantitative 869 intercomparison of the various statistical parameters. 870

871 β (high

freq (4-10 years)-1)

β (Lower freq than (25 years)-1)

Hhigh (4-10 years)

Hlow (>25 years)

C1

(Jones et al., 1998)

0.52 0.99 -0.27 0.063 0.104

(Mann et al., 1998), (Mann et al., 1999)

0.57 0.53 -0.22 -0.13 0.100

(Crowley and Lowery, 2000)

2.28 1.61 0.72 0.31 0.105

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(Briffa et al., 2001)

1.19 1.18 0.15 0.13 0.092

(Esper et al., 2002)

0.88 1.36 0.01 0.22 0.092

(Huang, 2004)

0.94 2.08 0.02 0.61 0.090

(Moberg et al., 2005)

1.15 1.56 0.09 0.32 0.094

(Ljundqvist, 2010)

_ 1.84 _ 0.53 0.098

872 Table 2: A comparison of parameters estimated from the multiproxy data from 1500-1979 (480 873 years). The Ljundquist high frequency numbers are not given since the series has decadal 874 resolution. Note that the β for several of these series was estimated in (Rybski et al., 2006) but 875 no distinction was made between low frequency weather and climate, the entire series were used 876 to estimate single (hence generally lower) β’s. 877

878 Fig. 9: The RMS Haar fluctuation for the mean of the pre and post 2003 series from 1500 – 879 1979 (solid blue and purple respectively and excluding the Crowley series due to its poor 880 resolution), along with the mean of the globally averaged monthly resolution surface series from 881 fig. 5 in (solid) red. In order to assess the effect of the 20th C warming, the structure functions for 882 the multiproxy data were recalculated from 1500 – 1900 only (dashed lines) and for the 883 instrumental surface series with their linear trends from 1880 -2008 removed (the data from 884 1880-1899 is too short to yield a meaningful S(Δt) estimate for the lower frequencies of interest). 885 While the large Δt variability is reduced a little, the basic power law trend is robust especially for 886 the post 2003 reconstructions. Note that the decrease in S(Δt) for the linearly detrended surface 887 series over the last factor of 2 or so in lag Δt is a pure artifact of the detrending. We may 888 conclude that the low frequency rise is not an artifact of an external linear trend. Reference lines 889 corresponding to β = 0.8 and 1.8 have been added. 890

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891 Fig. 8 compares the mean multiproxies with the ensemble average of the instrumental 892

global surface series. This confirms the basic behaviour: small Δt scaling with ξ(2)/2 = -0.1 (β = 893 0.8) followed by large Δt scaling with ξ(2)/2 = 0.4 (β = 1.8) is displayed by all the data, all the 894 pre-2003 S(Δt) functions drop off for Δt ≈>200 yrs. Notable are a) the transition scale in the 895 global instrumental temperature at τc ≈ 10 yrs which is somewhat lower than that found in the 896 reconstructions (τc ≈ 40 -100 yrs) and, b) over the common low frequency part, that the 897 amplitudes of the reconstruction RMS fluctuations are about a factor of two lower than for the 898 global instrumental series. The reason for the amplitude difference is not at all clear since on the 899 one hand, the monthly and annually averaged Haar structure functions of the instrumental series 900 are very close to each other up to Δt ≈ 10 yrs (the temporal resolution is not an issue) and 901 similarly, the difference between the northern and the southern hemisphere instrumental S(Δt) 902 functions is much smaller than this (only about ≈15%). 903

To get another perspective on this low frequency variability we can compare the 904 instrumental, and multiproxy structure functions with those from ice core paleo temperature 905 discussed in more detail in the next section. In fig. 10, we have superposed the calibrated 906 deuterium based RMS temperature fluctuations with RMS multiproxy and RMS surface series 907 fluctuations (we return to this in section 5). We see that extrapolating the latter out to 30 – 50 908 kyrs is quite compatible with the Vostok core with the “interglacial window” (i.e. the rough quasi 909 period and amplitude corresponding to the interglacials). Although the Vostok S(Δt) curve is 910 from the entire 420 kyr record (not just the Holocene), this certainly makes it plausible that while 911 the surface series appear to have realistic low frequency variability, that the variability of the 912 reconstructions is too small (although the post-2003 reconstructions are indeed more realistic 913 than the contrasting relative lack of variability in the pre -2003 reconstructions). 914

915 Fig. 10: This shows the RMS Haar fluctuations for the mean monthly, global surface 916

series (orange, left, from fig. 5), the mean pre-2003 and mean post-2003 proxies (red, dark blue; 917 bottom and middle left respectively, from fig. 9) as well as the mean Vostok S(Δt) function over 918

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the last 420 kyrs interpolated to 300 yr resolution and using the (Petit et al., 1999) calibration. 919 Also shown is the “interglacial window”, the probable typical range of fluctuations and quasi 920 periods of the glacial-interglacials. The “calibration” of the fluctuation amplitudes is accurate to 921 ±25%. 922

4.4 20th Century Exceptionalism 923

Although the reconstructions and instrumental series qualitatively agree, their 924 quantitative disagreement is large and requires explanation. In this section we consider whether 925 this could be a consequence of the 20th century “exceptionalism” discussed earlier, the fact that -926 irrespective of the cause – the 20th century is somewhat warmer than the 19th and earlier 927 centuries. It has been recognized that this warming causes problems for the calibration of the 928 proxies (e.g. (Ljundqvist, 2010)), and it will clearly contribute to the RMS multicennial 929 variability in fig. 9. In order to demonstrate that the basic type of statistical variability is not an 930 artifact of the inclusion of exceptional 20th C temperatures in fig. 9, we also show the 931 corrresponding Haar structure functions for the earlier period 1500-1900. Truncating the 932 instrumental series at 1900 would result in a series only 20 years long, therefore the closest 933 equivalent for the surface series was to remove overall linear trends, and then redo the analysis. 934 As expected the figure shows that all the large Δt fluctuations are reduced, but that the basic 935 scaling behaviours are apparently not affected. We conclude that both the type of variability as 936 characterized by the scaling exponents and the transition scale τc are fairly robust – if difficult to 937 accurately determine – they are not artifacts of external linear trends. The instrumental and 938 reconstruction discrepancy in fig. 9 thus remains unexplained. 939

5. Temporal spectral scaling in the climate regime: 10 - 105 yrs 940

5.1 Review of literature 941

Thanks to several ambitious international projects, many ice cores exist, particularly from 942 the Greenland and Antarctic ice sheets which provide surrogate temperatures based on δ18O or 943 deuterium concentrations in the ice. The most famous cores are probably the GRIP (Greenland 944 Ice core Project) and Vostok (Antarctica) cores, each of which are over 3 km long (limited by the 945 underlying bedrock) and go back respectively 240 and 420 kyrs. Near the top of the cores, 946 individual annual cycles can be discerned (in some cases going back over 10 kyrs); below that 947 the shearing of ice layers and diffusion between the increasingly thin annual layers makes such 948 direct dating impossible, and models of the ice flow and compression are required. Various 949 “markers” (such as dust layers from volcanic eruptions) are also used to help fix the core 950 chronologies. 951

A problem with the surrogates is their highly variable temporal resolutions combined 952 with strong depth dependences. For example, (Witt and Schumann, 2005) used wavelets, 953 (Davidsen and Griffin, 2010) used (monofractal) fractional Brownian Motion as a model, and 954 (Karimova et al., 2007) used (mono) fractal interpolation to attempt to handle this, (Lovejoy 955 and Schertzer, 2011) found that the temporal resolution itself has multifractal intermittency. 956 The main consequence is that the intermittency of the interpolated surrogates is a bit too high but 957

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that serious spectral biases are only present at scales of the order of the mean resolution or higher 958 frequencies. 959

With these caveats table 3 summarizes some of the spectral scaling exponents, scaling 960 ranges. It is interesting to note that the three main orbital (“Milankovitch”) forcings at 19 kyr, 23 961 kyr (precessional) and 41 kyr (obliquity) are indeed visible – but only barely - above the scaling 962 “background”, see especially (Wunsch, 2003). 963

964 Series Authors Series length

(kyrs) Reso-lution (yrs)

β c

Composite ice cores, instrumental

(Lovejoy and Schertzer, 1986)

Composite: Minutes to 106 years

1000 1.8

Composite (Vostok) (ice core, instrumental)

(Pelletier, 1998) 10-5 to 1000 0.1 to 500 2

δ18O from GRIP Greenland (Wunsch, 2003) 100 100 1.8 Planktonic δ18O ODP677 (Panama basin)

(Wunsch, 2003) 1000 300 2.3

CO2, Vostok (Antarctica) (Wunsch, 2003) 420 300 1.5 δ 18O from GRIP Greenland (Ditlevsen et al., 1996) 91 5 1.6 δ 18O from GRIP Greenland (Schmitt et al., 1995) 123 200 1.4 δ18O from GISP Greenland (Ashkenazy et al., 2003) 110 100 1.3 δ18O from GRIP Greenland (Ashkenazy et al., 2003) 225 100 1.4 δ18O from Taylor (Antarctica) (Ashkenazy et al., 2003) 103 100 1.8 δ18O from Vostok (Ashkenazy et al., 2003) 420 100 2.1 Composite, Midlatitude (Huybers and Curry,

2006a) 10-4 to 1000 0.1 to 103 1.6

Composite tropics (Huybers and Curry, 2006a)

10-4 to 1000 0.1 to 103 1.3

δ18O from GRIP Greenland (Lovejoy and Schertzer, 2011)

91 5 1.4

δ18O from Vostok (Lovejoy and Schertzer, 2011)

420 300 1.7

δ18O from GRIP, Greenland (Blender et al., 2006) 3 3 0.4 δ18O from GISP2 Greenland (Blender et al., 2006) 3 3 0.7 δ18O from GRIP Greenland (last 10 kyr only)

Fig. 8 10 5 0.2

Table 3: An intercomparison of various estimates of the spectral exponents βc of the low 965 frequency climate regime and scaling range limits. For series with resolution ≈ 100 yrs, the last 966 three rows are for the (anomalous) Holocene only, see (Lovejoy and Schertzer, 2011). 967

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5.2 A composite picture of atmospheric variability from 6 hours to 105 968

years 969

To clarify our ideas about the variability, it is useful to combine data over huge ranges of 970 scale into a single composite analysis (such as the spectra shown in fig. 1a, b) but using real 971 space fluctuations rather than spectra. Some time ago, such a composite already clarified the 972 following points: a) the distinction between the variability of regional and global scale 973 temperatures b) that the global averages had particularly small transition scale τc, c) that there 974 was a scaling range for global averages between scales of about 3 years and 40 - 50 kyrs (where 975 the variability apparently “saturates”) with a realistic exponent βc ≈ 1.8, d) that this scaling 976 regime could potentially quantitatively explain the magnitudes of the temperature swings 977 between interglacials (Lovejoy and Schertzer, 1986). 978

Similar scaling composites but in Fourier space were adopted by (Pelletier, 1998) and 979 more recently, by (Huybers and Curry, 2006a) who made a more data intensive study of the 980 scaling of many different types of paleotemperatures collectively spanning the range of about 1 981 month to nearly 106 years. The results are qualitatively very similar - including the positions of 982 the scale breaks; the main innovations are a) the increased precision on the β estimates and b) the 983 basic distinction made between continental and oceanic spectra including their exponents. We 984 could also mention the composite of (Fraedrich et al., 2009) which is a modest adaptation of 985 that of (Mitchell, 1976) although it does introduce a single scaling regime spanning only two 986 orders of magnitude: from ≈ 3 to ≈ 100 yrs (with β ≈ 0.3), although at lower frequencies the 987 composite exhibits a decrease (rather than increase) in variability. 988

Fig. 11 shows an updated composite where we have combined the 20CR reanalysis 989 spectra (both local, single grid point and global) with the GRIP 55 cm and GRIP high resolution 990 spectra (both for the last 10 kyr and averaged over the last 90 kyrs), the three surface global 991 temperature series. For reference, we have also included the 500 yr control run of the Institut 992 Pierre Simon Laplace (IPSL) GCM used in the 3rd IPCC assessment report. We use difference 993 structure functions so that the interpretation is particularly simple, although a consequence (see 994 section 2) is that all the logarithmic slopes are >0. In order avoid this problem, compare this to 995 the Haar structure functions, fig. 12. 996

Key points to note are a) the use of annually averaged instrumental data in fig. 11 997 (differences), but of daily data in fig. 12 (Haar), and b) the distinction made between globally 998 and locally averaged quantities whose low frequency weather have different scaling exponents. 999 Also shown is the interglacial window (Δt is the half quasi- period, and for a white noise, S is 1000 double the amplitude). The calibration of the paleotemperatures is thus constrained so that it 1001 goes through the window at large Δt but joins up to the local instrumental S(Δt) at small Δt. In 1002 addition, as discussed in section 4.2, since the last 10 kyr GRIP fluctuations are anomalously low 1003 (fig. 4 and see the nearly flat red curve compared with the full 91 kyr red curve); the calibration 1004 must be based on this flatter S(Δt) (fig. 11). Starting at τc ≈ 10 yrs, one can plausibly extrapolate 1005 the global S(Δt)’s using H = 0.4 (β ≈1.8), all the way to the interglacial window (with nearly an 1006 identical S(Δt) as in (Lovejoy and Schertzer, 1986)), although the northern hemisphere 1007 reconstructions don’t extrapolate as well, possibly because of their higher intermittency. The 1008 local temperatures extrapolate (starting at τc ≈ 20 yrs) with a lower exponent corresponding to β 1009

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≈1.4 (see fig. 1a) which is close to the other Greenland paleotemperature exponents (table 2 a) 1010 presumably reflecting the fact that the Antarctic temperatures are better surrogates for global 1011 rather than local temperatures, these exponents are all averages over spectra of series of ≈100 kyr 1012 or more in length. An interesting feature of the Haar structure function (fig. 12) is that it shows 1013 that local (grid scale) temperature fluctuations are roughly the same amplitude for monthly as for 1014 the much longer the glacial/interglacials periods. Not only can we make out the three scaling 1015 regimes discussed above, but for Δt> 100 kyrs, we can start to discern a new low frequency 1016 climate regime. For more scaling paleoclimate analyses, including “paleo-cascades”; see 1017 (Lovejoy and Schertzer, 2011). 1018

1019 Fig. 11: A comparison of the RMS structure function S(Δt) of the high resolution (5.2 yr) 1020

GRIP (red), IPSL (blue, 75oN and global), 20CR (gold, 75oN and global) mean surface series 1021 (purple), mean of the three post 2003 Northern hemisphere reconstructions (green) for globally 1022 averaged temperatures (bottom left set) and the mean at Greenland latitudes (upper set) all using 1023 fluctuations defined as differences (poor man’s wavelet) so that the vertical scale directly 1024 indicates typical changes in temperature. In addition, the GRIP data is divided into two groups: 1025 the Holocene (taken as the last 10 kyrs, lower red) and the entire 91 kyr of the high resolution 1026 GRIP series (upper red). The GRIP δ18O data have been calibrated by lining up the Holocene 1027 structure function with the mean 75oN 20CR reanalysis structure function (corresponding to 1028 ≈0.65 K/mil). When this is done, the 20CR and surface mean global structure functions can be 1029 extrapolated with exponent H ≈0.4 (see the corresponding line) to the “interglacial window” (red 1030 box) corresponding to half pseudo-periods between 30 and 50 kyrs with variations (= ±S/2) 1031 between ±2 and ±3 K. This line corresponds to spectral exponent β =1.8. Finally, we show a 1032 line with slope ξ(2)/2 =0.2 corresponding to the GRIP β =1.4 (see fig. 1 a, b); we can see that 1033 extrapolating it to 50 kyrs explains the local temperature spectra quite well. 1034

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1035 Fig. 12: This is the equivalent of fig. 11 except for the first order Haar structure function 1036

rather than the RMS difference structure function and including daily resolution 20CR data and 1037 monthly resolution surface temepratures. On the left top we show grid point scale (2ox2o) daily 1038 scale fluctuations for both 75oN and globally averaged along with reference slope ξ(1) = H = -0.4 1039 (20CR, 700 mb). On the lower left, we see at daily resolution, the corresponding globally 1040 averaged structure function. Also shown are the average of the three in situ surface series (fig. 5) 1041 as well as the post 2003 multiproxy structure function (from fig. 9). At the right we show both 1042 the GRIP (55 cm resolution, calibrated with 0.5 K/mil) and the Vostok paleotemperature series. 1043 All the fluctuations have been multiplied by 2.2 so that the calibration scale in degrees K is fairly 1044 accurate (compare with fig. 11). Also shown is the interglacial window and a reference line with 1045 slope -0.5 corresponding to Gaussian white noise. 1046

6. Conclusions: 1047

Just as the laws of continuum mechanics emerge from those of statistical mechanics 1048 when large enough numbers of particles are present, so do the laws of turbulence emerge at high 1049 enough Reynold’s numbers and at strong enough nonlinearity. However, the classical turbulence 1050 laws were constrained by strong assumptions of homogeneity and isotropy and could not cover 1051 wide scale ranges in the atmosphere. By generalizing them to take into account anisotropy 1052 (especially vertical stratification) and intermittency, their range of applicability is vastly 1053 increased. In the last five years - thanks in part to the ready availability of huge global scale 1054 data sets of all kinds - it has been possible to verify that these generalized emergent laws 1055 accurately hold up to planetary scales. For a recent review see (Lovejoy and Schertzer, 2010). 1056

These “weather regime” laws show that the horizontal variability is fundamentally 1057 controlled by solar forcing via the energy flux. First principles calculations show that this 1058

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accurately accounts for the large scale winds and predicts a drastic “dimensional transition” at τw 1059 ≈10 days, the typical lifetime of planetary scale structures. Beyond this time scale, spatial 1060 interactions are rapidly quenched so that the longer scales are driven by essentially temporal 1061 degrees of freedom and the spectra of atmospheric fields display a shallow “spectral plateau” 1062 with small logarithmic slope. By making a third generalization of the classical laws, the 1063 statistical behaviour in this “low frequency weather” regime can be predicted. This qualitative 1064 change in the statistics at τc is just as expected, it is not due to the action of new mechanisms of 1065 internal variability nor to external climate forcing, it is apparently nothing more than low 1066 frequency weather. Similarly, (forcing free) GCM control runs reproduce the same type of 1067 variability out to their low frequency limits. The main complication is that - due to similar 1068 effects from ocean turbulence whose corresponding outer time scale τo ≈ 1 year – there is 1069 enhanced intermittency up to that scale, with a slightly steeper “low frequency ocean weather” 1070 regime beyond. Depending on the location and atmospheric variable of interest, this scaling 1071 continues up to scales of ≈ 10 – 100 years beyond which the type of variability drastically 1072 changes; new mechanisms of internal variability or of external climate forcing must come into 1073 play: the true climate regime begins. It is notable that if we consider pre-20th century northern 1074 hemisphere temperatures, the result is qualitatively similar so that presumably anthropogenic 1075 forcings are not responsible for this new regime. 1076

In section 2, we discussed the weather and low frequency weather regimes using spectral 1077 techniques. However in later sections we turned to fluctuation statistics. For these the key 1078 parameter is the fluctuation exponent H which determines the rate at which the mean fluctuations 1079 increase (H>0) or decrease (H<0) with scale Δt. We underlined the particularly simple 1080 interpretations afforded by the usual difference and tendency structure functions. However, the 1081 corresponding wavelets only usefully characterize the scaling of the fluctuations over narrow 1082 ranges in the exponent H: the differences (0<H<1) and the tendency (-1<H<0), i.e. useful only 1083 for fluctuations increasing in amplitude with scale or decreasing in amplitude with scale 1084 respectively. Since both the weather and climate generally have H>0 and the low frequency 1085 weather regime has H<0, we instead defined fluctuations using the Haar wavelet which is thus 1086 useful over the entire range -1<H<1 and can also be “calibrated” to directly yield fluctuation 1087 amplitudes and can be easily implemented numerically. 1088

In order to evaluate the statistical variability of the atmosphere over as wide a range as 1089 possible, we combined the Haar wavelet with temperature data from the 20th Century reanalysis 1090 (1871-2008), (2ox2o, 6 hourly), three surface series (5ox5o, monthly), 8 intermediate length 1091 resolution “multiproxy” series of the Northern hemisphere from 1500- 1980 (yearly), and GRIP 1092 and Vostok paleotemperatures at 5.2 and ≈ 100 year resolutions over lengths 91 and 420 kyrs. 1093 The basic findings were that the key transition scale τc from low frequency weather to climate, 1094 was somewhat variable, depending on the field and geographical location. For example, for 1095 surface global temperatures we found τc ≈ 10 yrs whereas for the more reliable post 2003 1096 northern hemisphere reconstructions τc ≈ 30 yrs, for the Holocene GRIP (Greenland) core, τc ≈ 2 1097 kyrs, the Holocene Vostok (Antactica) core, τc ≈ 1 kyrs, and the mean pre Holocene 1098 paleotemperature value, τc ≈ 100 yrs. We also found Hlw ≈ -0.4, -0.1 for local (e.g. 2ox2o 1099 resolution) and global series respectively, and Hc ≈ 0.4 although the GRIP value was a little 1100 lower these values correspond to βlw ≈ 0.2, 0.8, 1.8 respectively (ignoring intermittency 1101 corrections K(2) – which ranged from Klw(2) ≈0.05 to Kc(2) ≈ 0.1 a full characterization of the 1102 intermittency i.e. K(q) was been performed, but was not discussed here). 1103

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Although this basic overall three-regime scaling picture is 25 years old, much has 1104 changed to make it more convincing. Obviously, an important element is the improvement in the 1105 quantity and quality of the data, but we have also benefited from advances in nonlinear dynamics 1106 as well as in data analysis techniques. In combination, these advances make the three-regime 1107 scaling model a seductive framework for understanding atmospheric variability over huge ranges 1108 of space-time scales. It allows us to finally clarify the distinction between weather, the 1109 straightforward extension of the weather dynamics - without new elements - to low frequency 1110 weather and finally to the climate regime. It allows for objective definitions of the weather 1111 (scales <τw), climate states (averages up to τc), and hence of climate change (scales >τc). This 1112 new understanding of atmospheric variability is essential in evaluating the realism of both 1113 atmospheric and climate models. In particular - since without special external forcing GCM’s 1114 only model low frequency weather - the question is posed as to what types of external forcing are 1115 required so that the GCM variability makes a transition to the climate regime with realistic 1116 scaling exponents and at a realistic time scales. 1117

7. Acknowledgements 1118

We thank P. Ditlevsen for providing the high resolution GRIP data, and A. Bunde for useful 1119 editorial and scientific comments. 1120

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