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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

The classical laws of turbulence associated with the names Kolmogorov, Corrsin, 8 Obhukov, and Bolgiano were expected to emerge when continuum mechanics was taken to the 9 strongly nonlinear limit. In order to apply to the atmosphere over wide ranges they must be 10 generalized to account for anisotropy and intermittency; over the last twenty five years this has 11 been done. We review the theory and compare it with recent global scale analyses showing that 12 the new laws hold up to planetary scales and account for weather dynamics up to time scales τw ≈ 13 10 days (the lifetime of planetary scale structures). We focus on the transition from the weather 14 to the climate at scales τ>τw. By making a third generalization of the classical laws, we show that 15 the spatial interactions become very weak causing a drastic “dimensional transition”: all spectra 16 become a relatively flat “spectral plateau”. The main complication is that high frequency “ocean 17 weather” undergoes the same transition but at scales τo ≈ 1 year. We show that the “weather-18 climate” regime beyond τw continues up to scales of 10 – 100 years giving way to the climate 19 regime proper which continues up to around 100 kyrs. Whereas the statistics of the weather-20 climate regime is accurately predicted as an extension of the basic weather regime, this new 21 climate regime involves new (but still scaling) physics and overall, there are three scaling 22 regimes. This allows us to objectively distinguish the weather from the climate on the basis of 23 its type of scaling variability. 24

25

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25

1. Introduction 26

1.1. What is the climate? 27

Notwithstanding the explosive growth of climate science over the last twenty years, there 28 is still no clear universally accepted definition of what the climate is – or what is almost the same 29 thing – what the difference is between the weather and the climate. The core idea shared by 30 most climate definitions is famously encapsulated in the dictum: “The climate is what you 31 expect, the weather is what you get” (Farmer’s Almanac). In more scientific language “Climate 32 in a narrow sense is usually defined as the "average weather," or more rigorously, as the 33 statistical description in terms of the mean and variability of relevant quantities over a period of 34 time ranging from months to thousands or millions of years (Intergovernmental Panel on Climate 35 Change. Appendix I: Glossary. Retrieved on 2007-06-01). 36

One problem with this definition is that it fundamentally depends on subjectively defined 37 space-time averaging scales. While the World Meteorological Organization defines climate as 38 30 year or longer variability, a period of two weeks to a month is often used to distinguish 39 weather from climate so that even with these essentially arbitrary periods, there is still a range of 40 about 1000 in scale (30 years /2 weeks) which is up in the air. This fuzzy distinction is also 41 reflected in numerical climate modelling since Global Climate Models are fundamentally the 42 same as weather models but at lower resolutions, with a different assortment of subgrid 43 “parametrisations” they are coupled to ocean models. Consequently, whether we define the 44 climate as the long term statistics of the weather, or in terms of the long term interactions of 45 components of the “climate system”, we still need an objective way to distinguish it from the 46 weather. 47

However, there is yet another problem with the Almanac and allied definitions: they 48 imply that climate dynamics are nothing new that they are simply weather dynamics at long time 49 scales. This seems naïve since we know from physics that when processes repeat over wide 50 enough ranges of space and or time scale we are used to observing that qualitatively new features 51 emerge so that over long enough time scales we expect that qualitatively new climate laws 52 should emerge from the higher frequency weather laws. These qualitatively new “emergent” 53 laws could simply be the consequences long range statistical dependencies of weather physics in 54 conjunction with qualitatively new climate processes; their nonlinear synergy giving rise to 55 emergent laws of climate dynamics. 56

Since the atmosphere is a nonlinear dynamical system with interactions and variability 57 occurring over huge ranges of space and time scales (millimetres to planet scales, milliseconds to 58 billions of years, ratios ≈1010 and ≈1020 respectively), the natural approach is to consider it as a 59 hierarchy of processes each with wide range scaling, i.e. each with nonlinear mechanisms that 60 repeat scale after scale over potentially wide ranges. Following (Lovejoy and Schertzer, 1986), 61 (Pelletier, 1998; Schmitt et al., 1995), (Koscielny-Bunde et al., 1998), (Talkner and Weber, 62 2000), (Blender and Fraedrich, 2003), (Ashkenazy et al., 2003), (Huybers and Curry, 2006b) 63 this approach is increasingly superceding earlier approaches that postulated more or less white 64 noise backgrounds with a large number of spectral “spikes” corresponding to many different 65 quasi-periodic processes. This includes the slightly more sophisticated variant (Mitchell, 1976) 66

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which retains the spikes but replaced the white noise with a hierarchy of Ornstein-Uhlenbeck 67 processes (white noises and their integrals); in the spectrum, “spikes” and “shelves”; see also 68 (Fraedrich et al., 2009) for a hybrid composite including a single short range scaling regime. 69

Over the past 25 years, scaling approaches have been increasingly applied to the 70 atmosphere, especially at meteorological scales, and in the last five years increasingly to global 71 scales. This has given rise to a new scaling synthesis covering the entire gamut of meteorological 72 scales from milliseconds to beyond the ≈10 day period which is the typical lifetime of planetary 73 structures i.e. the weather regime, reviewed in (Lovejoy and Schertzer, 2010b). It was 74 concluded that the theory and data were consistent with wide range but anisotropic spatial 75 scaling, and that the lifetime of planetary sized structures, provides the natural scale at which to 76 distinguish weather and climate. In this paper, we retain the same scaling framework, but 77 complement the earlier review by focusing on the longer time scales associated with the weather-78 climate transition and the climate proper. 79

The model discussed in this overview is the consequence of a series of generalizations of 80 the classical laws of turbulence associated with Kolmogorov, Corrsin, Obhukhov and Bolgiano. 81 After a quick empirical overview of the scaling using spectral analysis in space (section 2) and in 82 time (section 3), the classical laws are generalized in three important ways. The first is to 83 anisotropy (section 4) necessary at the very least to take into account the scaling vertical 84 stratification which is in turn a sine qua non for wide range horizontal and temporal scaling. The 85 second is the replacement of the classical picture of smooth, quasi Gaussian variability by strong 86 intermittency generated by multifractal cascades (section 5). These generalizations are enough 87 to model the atmosphere over “weather” scales (planetary scales in space and up to ≈ 10 days in 88 time). To go beyond this, we need a further generalization to cascade processes whose 89 nondimensional outer time scale is much longer than the corresponding (nondimensional) outer 90 space scale (section 6). This yields a drastic “dimensional transition”; followed by a 91 qualitatively different weather-climate regime which apparently holds up 30 -100 years. 92

2. A brief review of evidence for wide range spatial scaling using 93 spectra 94

2.1 Discussion 95

Although our focus is on time scales longer than the weather scales we first give a quick 96 empirical tour of the spatial, and then temporal scaling of the fields relevant either directly or 97 indirectly to atmospheric dynamics. This overview is by no means exhaustive; it partly reflects 98 the availability of relevant analyses and partly the significance of the fields in question. Spatial 99 scaling is fundamental for weather processes and it is the collapse (“dimensional transition”) of 100 the spatial interactions which distinguishes the high frequency weather from the low frequency 101 weather – climate regime out to 30 – 100 year scales. In order to simplify things as much as 102 possible, in this section we will only use spectra. 103

Consider a random field f(r) where r is a position vector (extensions to space-time are 104 straightforward). Its “spectral density” E(k) is the total contribution to the variance of the 105 process due to structures of with wavenumber between k and k + dk (i.e. due to structures of size 106 2π/l where l is the corresponding spatial scale. The spectral density thus satisfies: 107

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f r( )2

= E(k)dk0

!

" (1) 108

where f r( )

2

is the total variance (assumed to be independent of position; the angular brackets 109 “< . >” indicate statistical averaging). 110

In the following examples, we demonstrate the ubiquity of power law spectra: 111 E k( ) ! k"#

(2) 112

If we now consider the real space (isotropic) reduction in scale by factor λ we obtain: r! "#1r 113

corresponding to a “blow up” in wavenumbers: k! "k ; power law E(k) (eq. 2) maintain their 114 form under this transformation: E! "#$

E so that E is “scaling” and the (absolute) “spectral 115 slope” β is “scale invariant”. If empirically we find E of the form eq. 2, we take this as evidence 116 for the scaling of the field f. For the moment, we consider only scaling and scale invariance 117 under such conventional isotropic scale changes; in section 4 we extend this to anisotropic scale 118 changes. 119

2.2 The horizontal scaling of atmospheric Fields 120

The main data sources for horizontal scaling are remote sensing, aircraft, in situ networks 121 and reanalyses, each with their own particular advantages and disadvantages. We start by 122 considering global scale satellite radiances since they are the most straightforward to interpret. 123 Fig. 1 shows the “along track” 1-D spectra from the VIRS (Visible Infra Red Sounder) 124 instrument of the TRMM (Tropical Rainfall Measurement Mission) at wavelengths of 0.630, 125 1.60, 3.75, 10.8, 12.0 µm i.e. for visible, near infra red, and (the last two) thermal infra red. Each 126 channel was recorded at a nominal resolution of 2.2 km and was scanned over a “swath” 780 km 127 wide and ≈1000 orbits were used in the analysis. The scaling apparently continues from the 128 largest scales (20,000 km), to the smallest available. The scaling observed in the visible channel 129 (1) and the thermal IR channels (4, 5) are particularly significant since they are representative 130 respectively of the energy containing short and long wave radiation fields which dominate the 131 earth’s energy budget. Thanks to the effects of cloud modulation, the radiances are very 132 accurately scaling. This result is incompatible with classical turbulence cascade models which 133 assume well-defined energy flux sources and sinks with a source and sink - free “inertial” range 134 in between). Also of interest is the fact that the spectral slope β value is close (but a little lower) 135 than the value

! = 5 /3 expected for passive scalars in the classical Corrsin-Obukhov theory. 136 This result is consistent with theoretical studies of radiative transfer through passive scalar 137 multifractal clouds ((Watson et al., 2009), (Lovejoy et al., 2009c)). Although we cannot 138 directly interpret the radiance spectra in terms of the wind, humidity or other atmospheric fields, 139 they are strongly nonlinearly coupled to these fields so that the scaling of the radiances are prima 140 facie evidence for the scaling of the other fields. To put it the other way around: if the dynamics 141 was such that it primarily produced structures at a characteristic scale L, then it is hard to see 142 how this scale would not be clearly visible in the associated cloud radiances. 143

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144

Fig. 1: Spectra from ≈ 1000 orbits of the VIRS instrument (Visible Infrared Scanner) on the 145 TRMM satellite channels 1-5 (at wavelengths of 0.630, 1.60, 3.75, 10.8, 12.0 µm from top to 146 bottom, displaced in the vertical for clarity). The data are for the period January through March 147 1998 and have nominal resolutions of 2.2 km. The straight regression lines have spectral 148 exponents β = 1.35, 1.29, 1.41, 1.47, 1.49 respectively, close to the value β =1.53 corresponding 149 to the spectrum of passive scalars (= 5/3 minus intermittency corrections, see ch.3). The units 150 are such that k = 1 is the wavenumber corresponding to the size of the planet (20000 km)-1. 151 Channels 1, 2 are reflected solar radiation so that only the 15600 km sections of orbits with 152 maximum solar radiation were used. The high-wavenumber falloff is due to the finite resolution 153 of the instruments. Adapted from (Lovejoy et al., 2008b). 154

If we are interested in the usual meteorological variables of state rather than radiances we 155 may use reanalyses. Reanalyses are hybrid products in the sense that they combine data with 156 constraints imposed by the numerically discretized atmospheric equations in order to determine 157 an optimum estimate of the state of the atmosphere. Fig. 2 shows representative reanalyses taken 158 from the European Medium Range Weather Forecasting Centre (ECMWF) “interim” reanalysis 159 products, the zonal and meridional wind, the geopotential height, the specific humidity, the 160 temperature, vertical wind. They are publicly available at 1.5o resolution in the horizontal and at 161 37 constant pressure surfaces (every 25 mb in the lower atmosphere), every 6 hours from 1989 - 162 present. The data in fig. 2 were taken from the data rich 700 mb level. It also suffers little from 163 the extrapolations necessary to obtain global 1000 mb fields (which is especially problematic in 164 mountainous regions); see (Berrisford et al., 2009). 165

We used daily data for the year 2006 concentrating on the band between ±45o latitude 166 (with a cylindrical projection) allowing us to conveniently compare the statistics in the east-west 167 and north-south directions in order to study the statistical anisotropies between the two. The 168 east-west direction was broken up into 2 sections, one from 0-180o and the other from 180-0o 169 longitude. For technical reasons discussed in (Lovejoy and Schertzer, 2011), the spectrum was 170

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estimated by performing integrals around ellipses with aspect ratios 2:1 (EW:NS) the 171 wavenumber scale in fig. 2 indicates the east-west scale. 172

From the figure we can see excellent scaling except for generally small deviations at the 173 largest scales (≥ 5000 km), although for the geopotential, the deviations begin nearer to 2500 km. 174 While the scaling is quite convincing, the values of the exponents are not “classical” in the sense 175 that they do not correspond to the values predicted by any accepted turbulence theory. An 176 exception is the value β ≈ 1.6 for the humidity which is only a bit bigger than the Corrsin-177 Obukhov passive scalar value 5/3 (minus intermittency corrections which for this are of the order 178 of 0.15; see table 1, the intermittency parameters C1, α are described in section 5, the 179 nonconservation parameter H in section 4), although in any case classical (isotropic) turbulence 180 theory would certainly not be expected to apply at theses scales. We could also mention that 181 classically, the atmosphere is “thin” at these scales (since the horizontal resolution ≈ 166 km is 182 much greater than the exponential “scale height” ≈10 km), hence according to the classical 183 isotropic 3D/2D theory one would expect 2D isotropic turbulence to apply. This leads to the 184 predictions β = 3 for the horizontal wind field (a downscale enstrophy cascade) and β = 5/3 (with 185 an upscale energy cascade). In comparison, we see that the actual value for the zonal wind (β = 186 2.35) is in between the two. In (Lovejoy et al., 2009b) we argue that this is an artefact of using 187 gradually sloping isobars (rather than isoheights) in a strongly anisotropic (stratified) turbulence. 188 These spectra already caution us that reanalyses must be validated through scale by scale 189 statistical intercomparisons with other data. 190

191

Fig. 2: Inter-comparisons of the spectra of different atmospheric fields from the ECMWF interim 192 reanalysis. Top (red) is the geopotential (β = 3.35), second from the top (green) is the zonal wind 193

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(β = 2.40), 3rd from the top (cyan) is the meridional wind (β = 2.40), 4th from the top (blue) is the 194 temperature (β = 2.40) 5th from the top (orange) is the vertical wind (β = 0.4), at the bottom, 195 (purple) is the specific humidity (β = 1.6). All are at 700 mb and between ±45o latitude, every 196 day in 2006 at GMT. The scale at the far left corresponds to 20000 km in the east-west direction, 197 at the far right to 660 km. Note that for these 2-D spectra, Gaussian white noise would yield β = 198 -1 (a positive slope = +1). 199 200

Finally, let us consider in situ aircraft data, fig. 3; we see that the scaling for the 201 temperature, humidity and potential temperature is excellent over the larger range 560 m to 1140 202 km. The data are from a scientific aircraft (Gulstream 4) and were taken at 1 Hz (≈ 280 m 203 resolution), the data analyzed in fig. 3 are averages over 24 roughly straight legs, each with 4000 204 points long (≈ 1120 km), see (Lovejoy et al., 2009b), (Lovejoy et al., 2010). 205

206

Fig. 3: Aircraft spectra of temperature (blue, bottom), humidity (red, middle), log potential 207 temperature (gold, top), reference lines β = 2. These are averages over 24 isobaric aircraft 208 “legs” near 200 mb taken over the Pacific Ocean during the Pacific Winter Storms 2004 209 experiment, the resolution was 280 m (Nyquist wavenumber = (560 m)-1. Adapted from 210 (Lovejoy et al., 2010). 211

212

2.3 The atmosphere in the vertical 213

In spite of the fact that gravity acts strongly at all scales, the classical theories of 214 atmospheric turbulence have all been quasi-isotropic in either two or three dimensions. While a 215 few models tentatively predict possible transitions in the horizontal (for example between k-5/3 216 and k-3 spectra for a 3D to 2D transition for the wind), in contrast, in the vertical any 3D/2D 217 “dimensional transition” would be even more drastic (Schertzer and Lovejoy, 1985b). This is 218 also true for passive scalars. It is therefore significant that wind spectra from relatively low 219 resolution (50 m) radiosondes are scaling over nearly the entire troposphere (up to 13.3 km) with 220 slope near (but a little larger) than that predicted by Bolgiano and Obukhov (11/5) (Lazarev et 221 al., 1994). Using structure function analyses (section 4), this result has been extended down 222

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from ≈ 100 m down to 5 m resolutions with data from state-of-the-art drop sondes in (Lovejoy et 223 al., 2007). The empirical vertical spectral slope value βv ≈ 2.4 is greater than the horizontal 224 spectral slope value βh ≈ 5/3; although it is not obvious, this implies in fact that the atmosphere is 225 more and more stratified at larger and larger scales. The critical exponent ratio (βh-1)/(βv-1) = 226 Hz is quite near the theoretical value 5/9 discussed in section 4, the “23/9D” model, so-called 227 because the volume of typical stratified structures increases as the horizontal scale to the power 228 23/9 (≈2.55). It is the fact that the vertical and horizontal scalings are different that allows the 229 horizontal scaling to operate over huge ranges. 230

2.4 The spatial spectral scaling of the atmospheric boundary 231

conditions 232

The basic equations of the atmosphere are scaling, indeed, they have recently been shown 233 theoretically to be scaling under anisotropic scale changes (Schertzer et al., 2010), so that 234 solutions can potentially also be scaling with different horizontal and vertical exponents. 235 However, for this to be true, the boundary conditions must be scaling. Perhaps the most 236 significant boundary condition is the topography, including its prime importance for surface 237 hydrology and oceanography, and therefore the interactions hydrosphere-atmosphere. Scaling in 238 topography has a long history, going back nearly 100 years to when (Perrin, 1913) argued that the 239 coast of Brittany was nondifferentiable. Later, (Steinhaus, 1954) speculated on the 240 nonintegrability of the river Vistula, and (Richardson, 1961) quantified both aspects using scaling 241 exponents and (Mandelbrot, 1967) interpreted the exponents in terms of fractal dimensions. 242 Indeed, scaling in the earth’s surface is so prevalent that there are entire scientific specializations 243 such as river hydrology and geomorphology which abound in scaling laws of all types (for a 244 review see (Rodriguez-Iturbe and Rinaldo, 1997). 245

The first spectrum of the topography was the very low resolution one computed in 246 (Venig-Meinesz, 1951) who already noted that it was nearly a power law with β ≈ 2. After this 247 pioneering work, Balmino et al. (1973) made similar analyzes on more modern data sets and 248 confirmed Vening Meinesz’s results. Bell (1975) followed, combining various data sets (including 249 those of abyssal hills) to produce a composite power spectrum that was scaling over approximately 250 4 orders of magnitude in scale (also with β ≈ 2). More recent spectral studies of bathymetry over 251 scale ranges from 0.1 km to 1000 km can be found in Berkson and Matthews (1983) (β ≈ 1.6−1.8), 252 Fox and Hayes (1985) (β ≈ 2.5), Gibert and Courtillot (1987) (β ≈ 2.1−2.3) and Balmino (1993) 253 (β ≈ 2). Using the modern ETOPO5 data set (earth’s topography including bathymetry) at 5 254 minutes of arc (roughly 10 km) along with other higher resolution but regional Digital Elevation 255 Models (DEM’s) (Gagnon et al., 2006) found that the spectrum follows a scaling form with β ≈ 256 2.1 down to at least ≈ 40 m in scale (see this reference also for multifractal analyses. 257

Another important atmospheric boundary is the ocean surface; in section 4.3 we discuss 258 the scaling of currents and sea surface temperatures (SST) which are also argued to be scaling up 259 to planetary scales. Other surface fields are also scaling over wide ranges for example, remotely 260 sensed vegetation and soil moisture indices (Lovejoy et al., 2007). 261

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3. Temporal scaling, weather and climate 262

3.1 Temporal spectral scaling in the weather regime 263

Temporal scaling and its limits is the main subject of this overview, let us consider 264 evidence for its ubiquity. One of the earliest was that of (Van der Hoven, 1957) which is at the 265 origin of the famous “meso-scale gap”, the supposedly energy-poor spectral region between 266 roughly 10-20 minutes and ≈ 4 days (ignoring the diurnal spike). Even until fairly recently, 267 textbooks regularly reproduced the spectrum (often redrawing it on different axes or introducing 268 other adaptations), citing it as convincing empirical justification for the neat separation between 269 low frequency isotropic 2D turbulence - identified with the weather - and high frequency 270 isotropic 3-D “turbulence”. If the gap were real, the turbulence would be no more than an 271 annoying source of perturbation to the (2-D) weather processes. 272

However, it was quickly and strongly criticized (e.g. (Goldman, 1968), (Pinus, 1968), 273 (Vinnichenko, 1969), (Vinnichenko and Dutton, 1969 ), (Robinson, 1971) and indirectly by 274 (Hwang, 1970)). For instance, on the basis of much more extensive measurements (Vinnichenko, 275 1969) commented that even if the mesoscale gap really existed, it could only be for less than 5% of 276 the time; he then went on to note that Van der Hoven’s spectrum was actually the superposition of 277 four spectra and that the extreme set of high frequency measurements were taken during a single 278 one hour long period during an episode of ‘near-hurricane’ conditions and these were entirely 279 responsible for the high frequency “bump”. 280

More modern temporal spectra are compatible with scaling from dissipation scales to ≈ 10 281 days. Numerous wind and temperature spectra now exist from milliseconds to hours and days 282 showing β ≈ 1.6, 1.8 for v, T respectively, some of this evidence is reviewed in (Lovejoy and 283 Schertzer, 2011). For longer periods fig. 4 shows daily station data analyzed up to 60 years with 284 absolute slopes β =1.6 indicated. This is roughly the same value of β as in the horizontal; it 285 corresponds to an isotropic (x, y, t) space, a hypothesis to which we return below. According to 286 these spectra, it is plausible that the scaling in the wind holds from dissipation scales out to scales 287 of ≈ 5 -10 days where we see a transition (fig. 4). This transition is essentially the same as the low 288 frequency “bump” observed by Van der Hoven; its appearance only differs because he used a 289 ωE(ω) rather than logE(ω) plot. 290 291 292

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293 Fig. 4: Wind spectra from 36 near complete 60 year series of daily data from the continental US 294 taken from the stations lying nearest to 2ox2o degree grid points from 30o-50 o N, 105 o to 71 o W. 295 The brown curve (lower at left) is the daily maximum wind speed and the green is daily average 296 (normalized so that the annual peaks coincide). The spectra have been also averaged in the 297 frequency domain; in bins logarithmically spaced, 10 per order of magnitude (except for the 298 lowest factor of 10 where no spectral averaging was performed). The reference lines have 299 absolute slopes β = 1 (low frequencies), β = 0.2 (the plateau), and β =1.6 (high frequencies). 300

301 The corresponding spectra for hourly surface temperatures, are shown in fig. 5, from daily 302

series over 60 years long, fig. 6 (see also fig. 12 from daily series 6 years long). In all cases 303 multiple series were averaged to reduce the noise. The reference lines have slopes corresponding 304 (roughly) to the horizontally predicted values for the temperature (i.e. β ≈1.8). Again, the horizontal 305 values are seen to work well over the entire range up to the low frequency transition which in this 306 case starts at around a 7 - 20 day scale (figs. 4, 5, 6, several figures in the next section and fig. 12). 307 308

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309 Fig. 5: This shows the scaling of hourly surface temperatures from 4 stations in the northwest 310 US, for 4 years (2005-2008) from the US Climate Reference Network. One can see that in spite 311 of the strong diurnal cycle (and harmonics), that the basic scaling extends to about 7 days. The 312 reference lines (with absolute slopes 0.2, 2 are theoretically motivated, see section 6. The 313 spectra of hourly surface temperature data from 4 nearly colinear stations running north west - 314 south east in the US (Lander, WY, Harrison NE, Whitman NE, Lincoln NE), from the US 315 Climate Reference Network, 2005-2008. The thick line is the spectrum of the periodically 316 detrended spectrum, averaged over logarithmically spaced bins. 317

318 Fig. 6: Mean spectrum of daily dew point temperature, Td, temperature, T and relative humidity, 319 h for 36, 33, 7 stations respectively, (numbers vary due to missing data) from stations with long 320 (60 year; 22200 days) records. The low frequency and the “plateau” reference lines have slopes 321 -1 and -0.2 respectively. The spectra were averaged over 1dBω bins (i.e. 10 per order of 322

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magnitude in frequency), every 2o from 30o-50o latitude, from -105o to -71o longitude. The high 323 frequency reference line has absolute slope β =2 close to the horizontal β value for the humidity 324 and temperature which are each about 1.8 - 1.9 (see table 1), and the plateau value is very close 325 to the theory value 0.2. Adapted from (Lovejoy and Schertzer, 2010b). 326 Exponent Source hs T u v w z

ECMWF 0.40 5/3 5/3 5/3 1.10 3.35 β

(time) In situ 2.20d 1.67 1.68 1.68 __ 3.00

ECMWF 1.90 2.40 2.40 2.40 0.40 3.35 β

(space) In situ 1.9 1.9 1.6 1.6 _ _

ECMWF 0.10 0.075 0.083 0.086 0.115 0.085 C1

(time) In situ 0.09 0.087 0.088 0.088 __ 0.085

ECMWF 0.10 0.072 0.082 0.085 0.115 0.083 C1

(space) In situ 0.09 0.11 0.09 0.09 _ _

ECMWF 1.77 1.90 1.85 1.85 1.92 1.90 α

(time) In situ 1.8 1.61 1.5 1.5 _ 1.7

ECMWF 1.77 1.90 1.85 1.85 1.92 1.90 α

(space) In situ 1.8 1.8 1.9 1.9 _ _

ECMWF -0.21 1/3 1/3 1/3 0.17 1.26 Η

(time) In situ 0.68 0.41 0.33 0.33 __ 1.07

ECMWF

(space)

0.54 0.77 0.77 0.78 -0.14 1.26 H

(space)

aircraft 0.51 0.50 1/3e 1/3 __ __

Table 1: Spatial and temporal Scaling exponents 327 These are from the ECMWF interim reanalysis and from various in situ estimates (for 328

space, principally aircraft, (Lovejoy et al., 2010)). The temporal exponents were taken from the 329 literature and when no other source was available, from daily station data that had been selected 330 for a climate study; see the footnotes for details. Note that the ECMWF estimates were made at 331 the (hyper) viscous dissipation scale whereas the in situ data were estimated in the scaling 332 regime. Recalling the discussion in section 5, the difference is a factor ≈2.07 for the velocity 333 with possibly similar factors for the other variables. Due to the narrow range of frequencies (the 334 Nyquist frequency (2 days)-1)) to about (5 days)-1 before the spectra begin to flatten due to the 335 transition to climate), direct estimates are not accurate and depend on the exact frequency range 336 used. The zonal spatial outer cascade scales Leff were all in the range 13000 – 20000 km (except 337 for z which was ≈ 60000 km), the meridional values were about a factor 1.6 smaller. The outer 338

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time scales τeff were in the range 30- 60 days with the exception of z which was ≈ 300 days. For 339 full details, see (Lovejoy and Schertzer, 2011). 340 341

3.2 Temporal spectral scaling in the weather- climate regime 342

Excluding the annual cycle, roughly flat low frequency spectra are qualitatively reproduced 343 in all the standard meteorological fields and the transition scale is relatively constant at typically 5 - 344 10 days. This regime was called the “spectral plateau” in (Lovejoy and Schertzer, 1986) to 345 underline its ubiquity and to distinguish it from the higher frequency weather regime. In section 6, 346 we shall see that the basic stochastic fractionally integrated flux turbulence model that reproduces 347 the weather statistics when extended into the lower frequency predicts a dimensional transition 348 leading to a spectral plateau with βwc ≈ 0.2-0.4 for essentially all the fields (at least those over land), 349 over the ocean due to ocean turbulence it can yield βwc ≈ 0.6, hence in the figures discussed below 350 we also systematically show theoretically predicted reference lines with these slopes. Finally, 351 temporal spectra of the ECMWF 700 mb daily reanalysis data set for 2006 for T, u, v, hs, w and z 352 are shown in fig. 7. The overall shape and the transition scales are about the same as for the 353 instrumental series, although the exponents are not necessarily the same; a detailed intercomparison 354 is given in table 1. 355

356

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356

357 Fig. 7: The temporal spectra of the daily 700 mb ECMWF interim reanalysis fields that were 358 analyzed spatially in fig. 2 (i.e. between ±45o latitude). The dashed slopes are 5/3, the frequency 359 spectrum is shown estimated using ensemble and spectral averaging, units of ω: cycles/year. 360 The dashed lines have slopes -5/3, the solid lines have slopes -3.35, -0.4, -2.4, -1.1 (top to 361 bottom); they are drawn for ω > (11 days)-1. These correspond to the spatial z exponent, the 362 spatial w exponent (which accurately fits hs), the spatial u, v, T exponent and the regression w 363 slope respectively. The curves top (on left) to bottom are z, hs (multiplied by 10 from spatial 364 analysis, i.e. the spectra are multiplied by 100), v, T, u and w respectively. Note that the low 365 frequency rise is due to only two frequencies (1 yr)-1 and (6 months)-1, it is a low resolution 366 artefact of the annual cycle and subharmonic, not a true break in the scaling (see e.g. figs. 5, 8). 367 Reproduced from (Lovejoy and Schertzer, 2010a). 368 369

Moving to lower frequencies, we can use the longest existing reanalysis, the 20th 370 Century (20CR) reanalysis (Compo et al., 2009) from 1871- 2008. A particularity of the 20CR 371 is that it is based purely on surface pressure and monthly SST measurements. Here we use the 372 20CR data to compare different state variables at different locations, in particular at latitudes 373 which are mostly continental or mostly maritime in character. Figures 8 a, b which show the 374 spectra for T, u, v, hs from two 2o wide latitude bands, one tropical (centred on 5oN, mostly 375 maritime), the other midlatitude centred on 45oN (mostly continental; both from the 20CR 376 reanalysis product). We see that following the analysis of the temperature, that for the wind and 377 humidity, the 45oN weather-climate plateau (fig. 8 a) follows the continental βwc ≈0.2 prediction 378 very well up to around 10 years whereas the 5oN spectrum (fig. 8 b) is closer to the maritime 379 value βwc ≈ 0.6 (with the notable exception of the meridional wind which displays a βwc ≈0.2 – 380 0.4 region as indicated). Overall, we conclude that all the data can be reasonably modelled by 381 one or the other (βwc ≈0.2 or βwc ≈0.6) regimes being dominant. 382

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383 Fig. 8 a: Spectra from the 20CR reanalysis (1871-2008) at 45oN. The reference lines have 384 correspond to βοc = 0.6, βwc =0.2, βw =2 left to right respectively. 385

386 Fig. 8 b: Spectra from the 20CR reanalysis (1871-2008) at 5oN. The reference lines correspond 387 to βoc = 0.6, βwc =0.2, βw =2 left to right respectively. 388

389 Our basic empirical findings are more or less in accord with a growing literature – 390

particularly with respect to the temperature statistics. Our conclusions are especially close to 391 those of (Huybers and Curry, 2006a) who studied many paleoclimate series as well as the 60 392 year long NCEP reanalyses and concluded that for periods of months up to about 50 years, the 393 spectra are scaling with midlatitude βwc’s larger than the tropical βwc; (their values are 0.37±0.05, 394 0.56±0.08; our quite similar values of 0.2 – 0.4 and 0.6 work reasonably well, although as 395 explained in section 4, 6 it seems that the appropriate distinction is between continental versus 396

3/15/11 16

maritime regions). Using observations and climate models at monthly resolutions, (Fraedrich 397 and Blender, 2003), found βwc ≈ 0 over continents but βwc ≈ 0.3 in continental/ocean 398 “transitional regions”; a similar finding was made by (Huybers and Curry, 2006a) who also 399 noted a small mean land-sea difference (in their case of Δβwc =0.2); see table 2a. 400

(Bunde et al., 2004) gave evidence that over continents βwc was in the range 0.2 – 0.3 401 (similar to our estimates) and that the exponents implied the existence of long range correlations 402 in accord with our continental results. The stations analyzed by (Bunde et al., 2004) (see also 403 (Koscielny-Bunde et al., 1998) and (Lennartz and Bunde, 2009)) were primarily continental 404 and their exponents were fitted from 3 months to 10 years so that they do not resolve the 405 maritime versus continent issue (β ≈ 0.2 – 0.4 versus β ≈ 0.6). Recently, (Blender et al., 2006), 406 argued that at least in some regions such as the North Atlantic (but not Pacific), there is indeed a 407 long term memory (i.e. for Gaussian processes β ≠ 0) which at these scales, they attribute to the 408 thermal inertia of the Atlantic zonally averaged ocean circulation (see also (Fraedrich et al., 409 2009)). 410

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Series length (yrs)

βw τw (days)

βwc τc (yrs)

β c

Northern hemisphere instrumental (Lovejoy and Schertzer, 1986), 1 month

100 _ _ ≈0.2* 3 1.8

20th C (6 hours) (700 mb), global

138 3 10 ≈0.6* 5 ≈1.7*

NOAA NCDC, NASA GISS, HadCRUT3 (surface, appendix 10.C)

129 _ _ 0.2 (over land) 0.6 (SST)

10 1.7

Satellite global, (600 mb), daily

7 3 25 ≈0.2* 2 ≈2*

20thC (6 hours) (700 mb), 44N 138 2* 5 ≈0.2* 10 0.6* 20thC (6 hours) (700 mb), 5 N 138 2* 25 ≈0.6* _ 0.6* ECMWF interim (700 mb), year 2006

1 2* 5 ≈0.2, 0.6 _ _

Instrumental, daily, US 60 2* 7 ≈0.2* 10 1* (Lanfredi et al., 2009); deviations < 1 year considered as scale bound Markov process

≈100 0.26

(Huybers and Curry, 2006a) Midlatitude, tropics

Composite to 106

_ _ 0.4 (mid) 0.6 (tropics)

≈ 0.1 1.6 (mid) 1.3 (tropics)

(Fraedrich and Blender, 2003)

NCEP reanalyses, 60yrs, ECHAM4/ OPYC sims 500 yrs, in situ, 100 yrs

_ _ ≈0 (continents), ≈0.3 coasts

_ _

(Bunde et al., 2004; Koscielny-Bunde et al., 1998; Lennartz and Bunde, 2009),

In situ, 2 wks-30 yrs

_ 14 0.2 - 0.3 _ _

(Ditlevsen et al., 1996) Ice cores (GRIP) #

_ _ ≈ 0.2 3 0.8

(Lovejoy and Schertzer, 1986) Composite ice cores, instrumental

1.8 25 ≈ 0 3-300+

1.8

(Pelletier, 1998) Composite: ice core, instrumental

1.5 7 0.5 200 1.5

Table 2 a: An intercomparison of various estimates of spectral exponents β and scaling 411 range limits for the temperature. 412

The exponent βw is the high frequency weather regime exponent, τw is an estimate of the 413 scale where the spectral plateau begins. βwc is and estimate of the exponent in weather-climate 414 spectral plateau regime, and τc is a rough estimate of the long time (low frequency) limit to the 415 latter which has exponent βc. 416

*These values are not regression lines but from plausible reference lines indicated in the 417 figures; in most cases, the value β ≈0.3 would work nearly as well. 418 #Using monthly resolution GRIP ice core data for the last 3 kyr. 419

3/15/11 18

+ The 3 year figure was using northern hemisphere instrumental series, the 300 yr figure was 420 indirectly from Central England temperature series. 421

422 To complete this examination of the spectral plateau, let us consider a final relevant 423

climate scale variables (see table 2 a for a summary of temperature series characteristics, see 424 (Lovejoy and Schertzer, 2011) for further empirical analyses including of the North Atlantic 425 Oscillation index). In our scaling model outlined in more detail in section 4.1, it is the balance of 426 incoming and outgoing radiation (modulated by the clouds, abundances of H2O, aerosols, CO2, 427 O3, CH4, N2O (Tuck, 2008) and the surface variability all of which are scaling) that determine the 428 overall energy flux (ε) that drives the weather. Therefore, we expect ε to be particularly 429 significant for the climate. In fig. 9, we show an estimate from the 20CR reanalysis at 45oN 430 based of the Laplacian of the (dominant) zonal wind component. We see that it is similar to the 431 other 20CR fields although the plateau does indeed look very uniform and flat. 432

433 Fig. 9: The spectrum from 20CR reanalysis (1891-2002); the energy flux estimated from the 434 absolute Laplacian of the zonal wind at 700 mb, 42oN. The reference lines have βc = 2, βwc = 435 0.2, βw =1. Note the daily and annual spikes and subharmonics. 436 437 Series

length (yrs)

βw τw (days)

βwc τc (yrs)

β c

20CR Reanalysis

T 5o 138 1.7 25 0.6 10 0.6

T 45 o 138 3 4 0.2 10 0.6 u 5 o 138 3 5 0.6 10 0.6 u 45 o 138 2.6 4 0.2 10 0.6 v 5 o 138 3 5 0.2 10 0.6 v 45 o 138 3 4 0.2 10 0.6

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hs 5 o 138 3 5 0.6 10 0.6 hs 45 o 138 3 4 0.2 10 0.6 US instrumental

T 60 2 7 0.2 3 1.0

v 60 1.6 5 0.2 1 1.0 p 60 3 4 0.2 7 1.0 R 60 0.3 5 0.2 2 1.0 hs 60 2 7 0.2 7 1.0 T

(hourly) 4 2 5-7 0.2 _ _

Table 2b: An intercomparison of various spectral exponents and scaling range limits for 438 different atmospheric variables. 439

Data from the 20CR reanalysis and daily stations over the US. The values are not 440 regressions but reasonable reference lines, see the corresponding figures. The reanalysis 441 resolution is 6 hrs, the US instrumental resolution is 1 day (except as indicated). Given the 442 difficulty of estimating the exact limits of the scaling regimes and – for the low frequencies – the 443 noisy statistics - the exponents βwc, βw are not regression values but the exponents of reasonable 444 reference lines chosen from 0.2 (the “plateau cascade” value for βwc), 0.6 (the value of βwc 445 claimed for the tropics, globe) and 1 (the value that roughly corresponds to the qualitative 446 transition H =0). 447

3.3 Temporal spectral scaling of global temperatures: 448

The 20CR data has several interesting characteristics including its coverage of a range of 449 scales 6 hours to 138 yrs (a factor > 2x105), with no missing data and whose characteristics are 450 relatively homogeneous over time. However, it is based purely on surface pressure and monthly 451 sea surface temperature data, thus raising the question of how representative is it of the real 452 atmosphere. (Compo et al., 2009) give a partial answer by making specific comparisons with 453 the NCEP reanalysis products, but systematic statistical studies have not yet been made. In order 454 further answer to this question, we compare it to three surface temperature data sets. 455

The three we chose are the NOAA NCDC (National Climatic Data Center) merged land 456 air and sea surface temperature dataset (abbreviated NOAA NCDC below, from 1880 on a 5ox5o 457 grid, see (Smith, 2008) for details), the NASA GISS (Goddard Institute for Space Studies) data 458 set (from 1880 on a 2ox2o (Hansen et al., 2010) and the HadCRUT3 data set (from 1850 to 459 2010 on a 5o x5o grid). HadCRUT3 is a merged product created out of the HadSST2 (Rayner et 460 al., 2006) Sea Surface Temperature (SST) data set and its companion data set CRUTEM3 of 461 atmospheric temperatures over land. The NOAA NCDC and NASA GISS are both heavily 462 based on the Global Historical Climatology Network (Peterson and Vose, 1997), and have 463 many similarities including the use of sophisticated statistical methods to smooth and reduce 464 noise. In contrast, the HadCRUTM3 data is less processed. 465

Each pixel in each data set suffered from missing data points so that here we consider the 466 globally averaged series obtained by averaging over all the available data over the common 129 467 yrs period 1880 – 2008 (taking into account the latitude dependent map factors; fig. 10 shows the 468 spectra (see section 4.3 for analyses of the full 5o resolution data). These are quite similar 469 showing roughly βwc ≈ 0.6 up to ≈ 10 yrs, and for ω<(10 yrs)-1, roughly βc = 1.7, although clearly 470

3/15/11 20

there is too much scatter at these low frequencies for more precise exponent estimates. Analysis 471 of the scale by scale differences between the spectra is interesting, see (Lovejoy and Schertzer, 472 2011). In terms of the real space statistics, we can use the with the global root mean square 473 !T !t( )

21/2

annual structure functions (fitted for 129 yrs>Δt>10 yrs), obtaining 474 !T !t( )

21/2

≈0.08!t 0.33 for the ensemble, in comparison, (Lovejoy and Schertzer, 1986) found the 475

very similar !T !t( )21/2

≈ 0.077Δt0.4 using northern hemisphere data (these correspond to βc = 476 1.66, 1.8 respectively). 477

478 Fig. 10: The spectra (averaged over logarithmically spaced bins, 10 per order of magnitude, 479 using the same colours as previous). The units are such that ω = 1 corresponds to (129 yrs)-1; 480 note the annual spike, (1 year)-1 is at 2.11 on the log10ω axis). 481

3.4 Temporal spectral scaling in the climate regime proper: 10 - 105 482

yrs 483

In the last few chapters we have attempted to use a scaling framework combined with 484 instrumental data to understand atmospheric variability over time scales ranging from 485 milliseconds to 10 – 100 yrs; the weather regime followed by a broad weather-climate plateau. 486 In order to properly characterize the longer scale variations (and even to have more confidence in 487 our instrumental statistics from decades to longer periods), we need to use paleo climate data. 488 Since the paleo data we consider here are primarily ice core temperature proxies, it turns out that 489 scales 10 years and less are close to their maximum resolutions so that there is not as much 490 overlap with instrumental series as one might wish. By combining the smaller paleo scales and 491 the largest instrumental scales we can obtain an overall “composite” view of the variability over 492 the full range, and this in turn will help understand the weather-climate plateau. 493

Thanks to several ambitious international projects, many ice cores exist, particularly in 494 the Greenland and Antarctic ice caps which provide surrogate temperatures based on 18O or 495

3/15/11 21

deuterium concentrations in the ice. The most famous cores are probably the GRIP (Greenland 496 Ice core Project) and Vostok (antarctica) cores, each of which are over 3 km long (limited by the 497 underlying bedrock) and go back respectively 240 and 420 kyrs. Near the top of the cores, 498 individual annual cycles can be discerned (in some cases going back over 10,000 yrs); below that 499 the shearing of ice layers and diffusion between the increasingly thin annual layers makes such 500 direct dating impossible, and models of the ice flow and compression are required. Various 501 “markers” (such as dust layers from volcanic eruptions) are also used to help fix the core 502 chronologies. 503

A problem with the surrogates is their highly variable temporal resolutions combined 504 with strong depth dependences. For example, (Witt and Schumann, 2005) used wavelets, 505 (Davidsen and Griffin, 2010) used (monofractal) fractional Brownian Motion as a model, and 506 (Karimova et al., 2007) used (mono) fractal interpolation to attempt to handle this, (Lovejoy 507 and Schertzer, 2011) found that the temporal resolution itself has multifractal intermittency. 508 The main consequence is that the intermittency of the interpolated surrogates is a bit too high but 509 that serious spectral biases are only present at scales of the order of the mean resolution or higher 510 frequencies. 511

With these caveats table 3 summarizes some of the spectral scaling exponents, scaling 512 ranges. It is interesting to note that the three main astronomical (“Milankovitch”) forcings at 19, 513 23 kyr (precessional) and 41 kyr (obliquity) are indeed visible – but only barely - above the 514 scaling “background”. The predominance of the background is presumably associated with 515 nonlinear variability within the climate system and was noted from the 1980’s onwards. The 516 case for their relative dynamical insignificance was particularly strongly made by (Wunsch, 517 2003) who performed scaling analyses of various paleo spectra (see table 3), and who 518 specifically attempted to determine the portion of the variance which could be accounted for by 519 spectral bands near the three main Milankovitch frequencies. He concluded that even with a 520 liberal account of these bands, that they represent no more than 10% of the total. Wunsch went 521 on to propose various (nonscaling) Markov processes as models of the temperature. 522 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

3/15/11 22

Composite tropics (Huybers and Curry, 2006) 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)

(Lovejoy and Schertzer, 2011)

10 5 0.2

Table 3: An intercomparison of various estimates of the spectral exponents β c of the low 523 frequency climate regime and scaling range limits. 524

For series with resolution ≈ 100 yrs, the last three rows are for the (anomalous) Holocene 525 only, see (Lovejoy and Schertzer, 2011). 526

3.5 A composite temporal spectrum from 6 hours to 105 years 527

We have used a scaling framework and instrumental data in order to understand 528 atmospheric variability over time scales ranging from milliseconds to over a century; the weather 529 regime followed by a dimensional transition yielding a broad weather-climate plateau and then 530 the beginning of a new low frequency climate regime proper. With the help of paleotemperature 531 “surrogates”, we have extended this picture out to the limits of quantitative climatology i.e. to 532 105-106 yrs; we can now attempt to achieve a wide scale range composite of the overall 533 variability. 534

The best known – and apparently the first – attempt to produce a wide scale range picture 535 of atmospheric variability was (Mitchell, 1976). He produced what is still the most ambitious 536 single composite spectrum of atmospheric variability to date: ranging from hours to the age of 537 the earth (≈ 10-4 to 1010 yrs). Given the rudimentary quality of the data at that time, he admitted 538 that his composite was more of an “educated guess” than a quantitative spectrum. We already 539 mentioned that his model consisted of a juxtaposition of quasi-periodic processes (spectral 540 “spikes”) with Ornstein-Uhlenbeck processes which in the spectrum would give rise to a series 541 of “shelves”). 542

The first scaling composite model was (Lovejoy and Schertzer, 1986); which was based 543 on some of the early paleo temperatures from Greenland and Vostok ice cores, an ocean core 544 (although all were at the low resolutions available at that time) as well as local and hemispheric 545 instrumental temperatures at scales of centuries and less. Key features were a) the distinction 546 between the variability of regional and global scale temperatures b) finding that the global 547 averages had particularly long scaling regimes, c) evidence for a scaling range for global 548 averages between scales of about 3 years and 40 - 50 kyrs (where the variability apparently 549 “saturates”) with a climate exponent βc ≈ 1.8, d) the demonstration that this scaling regime could 550 quantitatively explain the magnitudes of the temperature swings between interglacials. 551

A similar scaling framework was also adopted by (Pelletier, 1998) and resulted in a very 552 similar composite; the main difference being the use of spectra rather than structure functions to 553 make the composite. More recently, (Huybers and Curry, 2006a) made a more data intensive 554 study of the scaling of many different types of paleotemperatures collectively spanning the range 555 of about 1 month to nearly 106 years. The result is qualitatively very similar to the previous 556

3/15/11 23

including the position of the scale breaks; the main innovations are a) the increased precision on 557 the β estimates and b) the basic distinction made between continental and oceanic spectra 558 including their exponents. We could also mention the composite of (Fraedrich et al., 2009) 559 which is a modest adaptation of that of (Mitchell, 1976) innovating by introducing a single 560 scaling regime spanning only two orders of magnitude: from ≈ 3 to ≈ 300 yrs (with β ≈ 0.3). 561

Fig. 11 shows our own updated composite where we have combined the 20CR reanalysis 562 spectra (both local, single pixel and hemispheric) with the GRIP 55 cm and GRIP high resolution 563 spectra. We could note that the use of the 700 mb reanalysis data is appropriate since this (rather 564 than the usual surface data) is roughly the level where the precipitation originates. We see that 565 the paleo and reanalysis spectra fit together very well. Since – unlike previous composites - 566 there is only a single “instrumental” source (the reanalysis) and single paleo calibration constant 567 for the GRIP δ18O series (which effectively allows all the paleo spectra to be simultaneously 568 moved up and down on the log-log plot), the results are relatively robust. From the figure we see 569 that whereas the local 75oN spectrum with βwc ≈ 0.2 overlaps well with the GRIP core out to its 570 limit ω = (138 yrs)-1, that the spectrum of the northern hemisphere mean temperature instead 571 falls on the extrapolation of the climate regime part of the GRIP spectra (with βwc ≈ 1.4). A 572 reference line with exponent βwc ≈ 0.6 - as suggested by the global scale and equatorial analyses 573 in fig. 8 – is also shown. This suggests a straightforward explanation for the difference between 574 the hemispheric and local temperatures: a climate process exists which varies the entire 575 hemispheric temperature and competes with the weather process. At weather scales, the climate 576 process has a very low amplitude, its variability (spectral power) is well below that of the 577 weather processes. However, since its exponent is much larger, at low enough frequencies it 578 becomes dominant. Starting at ω ≈ (10 yrs)-1 it first dominates the global scale average 579 temperature spectrum since this has a smaller amplitude; only later, (at around ω ≈ (300 yrs)-1) 580 does it dominate the more variable local temperature. For more scaling paleoclimate analyses, 581 including “paleo-cascades” (Lovejoy and Schertzer, 2011). 582

583 Fig. 11: A modern composite based purely on the GRIP core and 20CR reanalyses. All 584

spectra have been averaged over logarithmically spaced bins, 10 per order of magnitude. Spectra 585 of northern hemisphere temperatures (red; from the 20 CR reanalysis, 1871-2008), blue is the 586

3/15/11 24

single 2 ox2o) pixel spectrum at 75oN also from the same reanalysis (and at the same scale). The 587 light green is the mean of the GRIP high for last 90 kyrs and the (lowest) frequency blue is from 588 the GRIP 55 cm core interpolated to 200 yr resolution and going back 240 kyrs. The solid 589 reference lines have absolute slopes βwc = 0.2, βwc = 0.6 and βc = 1.4 and βw = 2 as indicated. 590 The dashed line has absolute slopes βwc = 0.6 which is suggested by the global scale and 591 equatorial analyses in fig. 8. The red arrows at the bottom (and upper right) indicate the basic 592 qualitatively different scaling regimes. Reproduced from (Lovejoy and Schertzer, 2011). 593

4. Anisotropic generalizations of the classical laws of turbulence and 594

the spectral plateau 595

4.1 Real and Fourier Space-time scale functions 596 We have argued that atmospheric variables including the wind have wide range 597

(anisotropic) scaling statistics and that spatial scaling of the horizontal wind leads to the temporal 598 scaling of all the fields. Unfortunately, space-time scaling is somewhat more complicated than 599 pure spatial scaling. At meteorological time scales this is because we must take into account the 600 mean advection of structures and the Galilean invariance of the dynamics. At longer (climate) 601 time scales, this is because we consider the statistics of many lifetimes (“eddy-turn-over times”) 602 of structures. We first consider the shorter timescales. This is a summary of the more detailed 603 discussion in (Lovejoy et al., 2008b), (Lovejoy and Schertzer, 2010b), (Lovejoy and 604 Schertzer, 2011). 605

In order to illustrate the formalism, consider the horizontal wind v. In the stratified 606 turbulence mentioned in section 2.3 (the 23/9D model), the energy flux ε dominates the 607 horizontal and the buoyancy variance flux φ dominates the vertical so that horizontal wind 608 differences follow: 609

!v !x( ) = "1/3!x1/3; a

!v !y( ) = "1/3!y1/3; b

!v !z( ) = #1/5!z3/5; c

!v !t( ) = "1/2!t1/2; d

(3) 610

where Δx, Δy , Δz, Δt are the increments in horizontal coordinates, vertical coordinate and time 611 respectively. Equations (3 a - b) describe the real space horizontal Kolmogorov scaling and 3 c 612 the vertical Bolgiano-Obukhov (BO) scaling for the velocity, the equality signs should be 613 understood in the sense that each side of the equation has the same scaling properties. The 614 anisotropic Corrsin-Obukov law for passive scalar advection is obtained by the replacements 615 v! "; # ! $ 3/2#%1/2 where ρ is the passive scalar density, χ is the passive scalar variance 616 flux. 617

We have included eq. 3 d which is the result for the pure time evolution in the absence of 618 an overall advection velocity; this is the classical Lagrangian version of the Kolmogorov law 619 (Inoue, 1951; Landau and Lifschitz, 1959), it is essentially the result of dimensional analysis 620 using ε and Δt rather than ε and Δx. Although Lagrangian statistics are notoriously difficult to 621 obtain empirically (see however (Seuront et al., 1996)), they are roughly known from experience 622

3/15/11 25

and are used as the basis for the space-time or “Stommel” diagrams that adorn introductory 623 meteorology textbooks (see (Schertzer et al., 1997), (Lovejoy et al., 2000) for scaling 624 adaptations). 625

The classical turbulence laws are isotropic whereas eq. 3a-d has different exponents in 626 the horizontal (x,y), vertical (z) and time (t). However, the statistics impied by eq. 3 a-d have no 627 characteristic lengths, they are anisotropic scaling generalizations of the classical laws. To see 628 this explicitly, we can express the scaling (eqs. 3 a -d) in a single expression valid for any space-629 time vector displacement !R = !r,!t( ) = !x,!y,!z,!t( ) by introducing a scalar function of 630 space-time vectors called the “(space-time) scale function”, denoted

!R! " , which satisfies the 631

fundamental (functional) scale equation: 632

!"Gst #R!"

#$ = !"1 #R% &; G

st=

Gs

0

0 Ht

$

%&&

'

()); H

t= 1 / 3( ) / 1 / 2( ) = 2 / 3 (4) 633

where Gs is the 3X3 matrix spatial generator: 634

Gs=

1 0 0

0 1 0

0 0 Hz

!

"

###

$

%

&&&; H

z= 1 / 3( ) / 3 / 5( ) = 5 / 9

(5) 635

(with rows corresponding to (x,y,z)) and Gst is the extension to space-time. The spatial 636 stratification model with Gs given in eq. 5 has structure whose volume are proportional to 637 L.L.LHz=LDel with Del =2+Hz =23/9=2.55… L is their horizontal extent, Del is the “elliptical 638 dimension”; this is the “23/9D turbulence model” (Schertzer and Lovejoy, 1985b). 639

Using the space-time scale function, we may now write the space-time generalization of 640 the Kolmogorov law (eq. 2) as: 641

!v !R( ) = "!R! "#

!R! "H

; H = 1 / 3; # = 1 / 3 (6) 642

where the subscripts on the flux indicate the space-time scale over which it is averaged. This 643 anisotropic generalization of the Kolmogorov law is thus one of the key emergent laws of 644 atmospheric dynamics and serves as a prototype for the emergent laws governing the other 645 fields. Equation 6 is the prototypical emergent turbulent law, it relates fluctuations in an 646 observable (Δv) to an underlying turbulent flux ε. In section 5 we generalize it further by 647 considering the cascade generation of highly intermittent (multifractal) fluxes. Note that 648 although we do not consider it here, off diagonal elements of G correspond to structures whose 649 orientation as well as “squashing” depends on scale, further non matrix (nonlinear) Generalised 650 Scale Invariance (Schertzer and Lovejoy, 1985a) is needed for structures whose morphologies 651 change with location as well as scale (such as clouds, e.g. (Lovejoy and Schertzer, 2011)). 652

The simplest (“canonical”) space-time scale function satisfying eq. 4 is: 653

!R! "can

= ls

!rls

"

#$%

&'

2

+!t(s

"

#$%

&'

2 /H("

#$

%

&'

1/2

(7) 654

3/15/11 26

Where ls= !"3/4# 5 /4 is the dimensionally unique length scale called the “sphero-scale” and 655

!s= "#1/2$1/2 is the corresponding “sphero-time” (the lifetime of structures of size equal to the 656

sphero-scale). At scales ls, structures are typically roundish (since 657

ls,0,0,0( )!

"#$ = 0,l

s,0,0( )!

"#$ = 0,0,l

s,0( )!

"#$ ), hence the name. With the scale function (eq. 7), the 658

fluctuations (eq. 6) respect eqs 3 a- d. 659 Using the Lagrangian temporal scaling (eq. 3 d) implies Hτ≠1 apparently predicting 660

different horizontal and temporal scalings; this is in contradiction with the empirical analyses of 661 the previous sections which showed that horizontal and temporal exponents were very close to 662 each other. However, we are interested in the Eulerian frame temporal scaling for this, we are 663 missing a key ingredient: advection. When studying laboratory turbulence generated by an 664 imposed flow with velocity V with superposed turbulent fluctuations, (Taylor, 1938) proposed 665 that the turbulence is “frozen” such that the pattern of turbulence blows past the measuring point 666 sufficiently fast so that it doesn’t have time to evolve; i.e. he proposed that the spatial statistics 667 could be obtained from time series by the deterministic transformation VΔt -> Δx where V is a 668 constant: in the lab it is determined by the fan and by the wind tunnel geometry. While this 669 transformation has been frequently been used in interpreting meteorological series, it can only be 670 properly justified by assuming the existence of a scale separation between small and large scales 671 so that the large scales really do blow the small scale (nearly “frozen”) structures past the 672 observing point. Since we have argued that there is no scale separation in the atmosphere this is 673 problematic. 674

As a first step in taking advection into account we can use the Gallilean transformation 675

r! r " vt , t! t . The second step is to consider v as a random vector; with mean vx,v

y( ) (for 676

simplicity we consider only the horizontal) and variance V2= v

x

2+ v

y

2

(in (Lovejoy and 677 Schertzer, 2010b; Lovejoy and Schertzer, 2011) we give the full details). If the scaling holds 678 to planetary scales, then this V is the typical velocity difference between structures at antipodes; 679 V = Vw = ε1/3Le

1/3 where the scale of the earth is Le =20000 km. (the subscript “w” is for 680 “weather”). From this, we can estimate the typical timescale τw = Le/Vw and use these to 681 nondimensionalize the coordinates (denoted by a circumflex “ ! ”). For more realism we can 682 introduce the meridional/zonal aspect ratio a to obtain: 683

!x! =!x

Lw

; !y! =!y

aLw

; !t" =!t

"w

; v# x = µx=vx

Vw

; v# y = µy=vy

Vw

(8) 684

the symbols µx, µy are used for the components of the nondimensional velocity; they are less 685

cumbersome than v! x , v!y where: 686

Vw= v

x

2+ a

2vy

2( )1/2

; !w=Lw

Vw (9) 687

It is now convenient to define: 688

3/15/11 27

µ = µx,µ

y( ) µ2

= µx

2 + µy

2

(10) 689

which satisfies µ < 1 . In terms of the nondimensional quantities this yields an “effective” 690 nondimensional scale function: 691

!R!"#$%&'eff

" !R!T

B!R!( )1/2

; B =

1 0 #µx

0 a2 #a2µy

#µx #a2µy 1

$

%

&&&&

'

(

))))

; !R! = !x!,!y!,!t(( )

(11) 692

where “T” indicates “transpose”. The scale function in eq. 11 is only “trivially anisotropic” since 693

it is scaling with respect to an “effective” G matrix Geff = 1= the identity; the matrix B

simply 694 determines the trivial space-time anisotropy. This means for example that aspect ratios of 695 structures are independent of scale (empirically, (Lovejoy and Schertzer, 2010a) find a ≈ 1.6 696 for the main state variables corresponding to a mean zonal “squashing” of structures, the 697 exception is the meridional velocity which – due to symmetry - has a ≈ 1/1.6 i.e. squashing in the 698 meridional direction). 699

Hence, in horizontal space-time we have: 700

!V !R( ) = "1/3

!RT!B!R"( )

1/6

(12) 701 In order to test out the correctness of the effective (horizontal-time) global space-time 702

scale function (eq. 12) it is convenient to use Fourier techniques. If we assume that the structure 703 function of a field I is scaling, then this implies the scaling of the spectral density (Pst in space 704 dimension Dst=2+1=3), with a (different) fourier space (represented by a tilde) scale function: 705

!I !R( )q

= !R! "" q( ); P

stK( ) = #I K( )

2

# K! "$ sst; s

st= D

st$ " 2( ) (13) 706

!I !R( )q

is the qth order structure function and ξ(q) is its scaling exponent. !I2

and 2(1-707 P) are Fourier transform pairs (this is because of the link between the second order structure 708 function, the autocorrelation function and the spectrum, see e.g. (Lovejoy and Schertzer, 709

2011)), K = (k,ω), k = (kx, ky) and !R! " satisfy the scaling equation with respect to Gst , K! " 710 with respect to GT

st and Dst, is the trace of the space-time Gst. The subscript “st” is for space-711

time” i.e. x, y, t space. If eq. 13 is regarded as defining the Fourier scale function K! " , then we 712

must have K! "≥0, (as for real –space scale functions), this is assured since Pst(K) is ≥0. We can 713 similarly define the purely spatial Fourier space scale function from the spatial (k) subspace of K. 714

With the corresponding nondimension fourier vector: K! = kx!,ky!,!!( ) = Lwkx ,Lwky ,"w!( ) 715

we obtain the general result: 716

3/15/11 28

717

!R!"#$%&' = !R!

T

B!R!( )1/2

; K("#$%&' = K(

T

B"1K(( )

1/2

; detB > 0

(14) 718

These scale functions satisfy eq. 13; eq. 14 is the relation between their respective “trivial” 719 anisotropies. In (Lovejoy and Schertzer, 2011) we shall see that the scale functions for non 720

(space-time) localized (wavelike) behaviour detB <0. More details and an empirical verification 721 of the above using geostationary MTSAT thermal IR data (including waves) is discussed in Pinel 722 et al 2011, (Lovejoy and Schertzer, 2011). 723

4.2 The weather-climate transition scale from “first principles” 724

We have shown evidence that temporal scaling holds from small scales to a transition 725 scale τw of around 5 -10 days. Let us now consider the physical origin of this scale. In the 726 original Van der Hoven ωE(ω) versus logω plot, it appeared as a low frequency spectral “bump” 727 and its origin was argued to be due to “migratory pressure systems of synoptic weather- map 728 scale” (Van der Hoven, 1957). The corresponding features at around 4 – 20 days notably for 729 temperature and pressure spectra were termed “synoptic maxima” by (Kolesnikov and Monin, 730 1965), and (Panofsky, 1969) in reference to the similar idea that it was associated with synoptic 731 scale weather dynamics see (Monin and Yaglom, 1975) for some other early references. 732

If there is (at least statistically) a well defined relation between spatial scale and lifetimes 733 (the “eddy-turn-over time”), then the lifetime of planetary scale structures is of fundamental 734 importance. Since the shorter periods τ<τw statistics are dominated by structures of planetary 735 size τ ≈ εl

-1/3l -2/3 <τw whereas for τ > τw, they are dominated by the statistics of many lifetimes of 736 planetary scale structures. If we evaluate the lifetime at planetary scales Le i.e. we assume that 737 the outer weather scale ≈ Le, and use eq. 3 a, b then we obtain a large scale velocity Vw ≈ 738 εw

1/3Le1/3 which is the typical velocity across a structure of size Le. The corresponding eddy-turn 739

over time/ lifetime of planetary structures is therefore τw = Le/Vw = εw−1/3 Le

2/3. 740 But what determines the globally averaged fundamental flux εw? We can estimate the 741

mean energy flux εw by using the fact that the mean solar flux absorbed by the earth is ≈200 742 W/m2 (e.g. (Monin, 1972)). If we distribute this over the troposphere (thickness ≈ 104 m), with 743 mean air density ≈ 0.75 Kg/m3, and we assume a 2% conversion of energy into kinetic energy 744 ((Palmén, 1959), (Monin, 1972)), then we obtain a value εw ≈ 5x10-4m2/s3 which is indeed 745 typical of the values measured in small scale turbulence (Brunt, 1939), (Monin, 1972). Using 746 the ECMWF interim reanalysis to obtain a modern estimate of εw (Lovejoy and Schertzer, 747 2010b) showed that although ε is larger in mid latitudes than at the equator and that 300 mb it 748 reaches a maximum, the global average is ≈ 10-3 m2/s3. They also showed that using latitudinally 749 varying ε, that this explained to better than ±20% the latitudinal variation of the hemispheric 750 antipodes velocity differences (using Δv = ε1/3Le

1/3) and concluded that the solar energy flux does 751 a good job of explaining the horizontal wind fluctuations up to planetary scales. 752

If we now assume that the horizontal dynamics are indeed dominated by the energy flux, 753 then we can use Kolmogov’s formula to extrapolate these first principles estimates up to 754

3/15/11 29

planetary scales to estimate the large scale velocity difference across a hemisphere and we obtain 755 Vw ≈ 17 m/s. The corresponding eddy-turn over time, lifetime, is therefore τw ≈ 10 days i.e. 756 roughly the time associated with synoptic – global scale phenomena, the transition to the spectral 757 plateau as discussed in section 3.4. 758

Although this “first principles” calculation of the weather velocity Vw and time scales τw 759 from the solar energy input is seductive, as far as we can tell, it was not proposed until recently 760 (Lovejoy and Schertzer, 2010b), presumably because the Kolmogorov law was believed to 761 only hold in its isotropic form so that the classical relation τ ≈ ε−1/3L2/3 couldn’t possibly apply to 762 such large scales. 763

4.3 Ocean “weather”, the ocean - climate transition and the ocean 764

spectral plateau 765

It is well known that ocean variability is important for atmospheric dynamics for months 766 and longer time scales; before explicitly attempting to extend our model of weather variability 767 beyond τw, we must first consider the role of the ocean. The ocean and the atmosphere have 768 many similarities; from the preceding discussion, we may expect analogous regimes of “ocean 769 weather” to be followed by an ocean spectral plateau both of which will influence the 770 atmosphere. To make this more plausible, recall that both the atmosphere and ocean are large 771 Reynolds’ number turbulent systems and both are highly stratified - albeit due to somewhat 772 different mechanisms. In particular there is no question that at least over some range that 773 horizontal ocean current spectra are dominated by the ocean energy flux εo and hence follows 774 roughly E(ω) ≈ω-5/3 (and presumably in the horizontal: E(k) ≈ k-5/3); see e.g. (Grant et al., 1962), 775 (Nakajima and Hayakawa, 1982). Although surprisingly few current spectra have been 776 published, of late the use of satellite altimeter data to estimate sea surface height (a pressure 777 proxy) has provided relevant empirical evidence and has somewhat revived the debate about the 778 spectral exponent and scaling of the current. At scales where approximate geostrophic 779 equilibrium may pertain, the pressure gradient is proportional to the current so that the surface 780 height and current spectral exponents are related by βh = 2+βv. According to (Le Traon et al., 781 2008), at least over the scale range accurately covered by altimeters (≈ 10- 300 km), βh ≈ 11/3 782 implying that βv ≈ 5/3 (in contradiction with the competing Quasi-Geostrophic prediction βh ≈ 5, 783 (Smith and Vallis, 2001)); for more details and review, see (Lovejoy and Schertzer, 2011). 784 The existing data are apparently compatible with k-5/3 horizontal current spectra out to planetary 785 scales, hence of the ocean energy flux εo is the dominant flux. Although empirically the current 786 spectra (or their proxies) at scales larger than several hundred kilometres are not well known, 787 other spectra, especially those of sea surface temperatures (SST) are known to be scaling over 788 wide ranges and are relevant due to their strong nonlinear coupling with the current. Using 789 mostly remotely sensed infra red radiances, and starting in the early 1970’s there is much 790 evidence for SST scaling up to thousands of kilometres with β ≈ 1.8 -2 i.e. nearly the same as for 791 the atmospheric temperature (see e.g. (McLeish, 1970), (Saunders, 1972), (Deschamps et 792 al., 1981), (Deschamps et al., 1984), (Burgert and Hsieh, 1989), (Seuront et al., 1996), 793 (Lovejoy et al., 2000), and a review in (Lovejoy and Schertzer, 2011)). 794

3/15/11 30

If, as in the atmosphere where the stratification is scaling, so it may well be in the ocean, 795 leading to the likelihood that - as in the atmosphere - the energy flux dominates the horizontal 796 ocean dynamics then we can use the same methodology as in the previous subsection -basic 797 turbulence theory (the Kolmogorov law) combined with the mean ocean energy flux εo - to 798 predict the outer scale τo of the ocean regime. Thus, for ocean gyres and eddies of size l, we 799 expect there to be a characteristic eddy turnover time (lifetime) τ = ε-1/3l2/3 with a critical “ocean-800 weather” – “ocean-climate” transition time scale τo = εo

-1/3Lo2/3 (Lo is the outer spatial scale of the 801

oceans; presumably Lo ≈ Le =2x107m). Again, we expect a fundamental difference in the 802 statistics for fluctuations of duration τ< τo - the ocean equivalent of “weather” with a turbulent 803 spectrum with roughly βo ≈ 5/3 (at least for the current) - and for durations τ > τo, the ocean 804 “climate” with a shallow ocean spectral plateau with β ≈ <1. 805

In order to test this idea, we need εo, the globally averaged ocean current energy flux. As 806 expected, εo is highly intermittent (see (Robert, 1976), (Clayson and Kantha, 1999), (Moum 807 et al., 1995), (Lien and DʼAsaro, 2006), (Matsuno et al., 2006)) and as far as we know, the 808 only attempt to estimate its global average is (Lovejoy and Schertzer, 2011) who argue that 809 ocean drifter maps of eddy kinetic energy can be used to estimate it. They found that εo ≈10-8 810 m2/s3 is a reasonable global estimate for the surface layer (it decreases quite rapidly with depth). 811 Using the formula τo = εo

-1/3Lo2/3 we find that a range of εο between 1x10-8 - 8x10-8 m2s-3 812

corresponds closely to the range of τo ≈ 1 - 2 years; as expected, this is somewhat larger than the 813 corresponding value for the atmosphere (εw ≈10-3m2s-3, τw ≈ 10 days). 814

815 This provides us with a prediction for the SST spectrum: E(ω)≈ ω-1.8 for ω ≈≥(1 year)-1 816

and then a transition to a much flatter plateau for the lower frequencies. A way to test the model 817 is to filter out high frequency weather variability due to the weather dynamics; this at least 818 partially done by temporal averaging over scales >τw. A one month resolution allows us to use 819 the data sets described in section 3.4. 820

For all data sets and for virtually any given pixel, the series had many missing months, 821 and these were often successive so that interpolation could easily lead to serious biases in the 822 spectra and moments. To minimize this problem, we restricted the separate land and SST 823 analyses to the most recent 100 years and selected only those pixels with less than 20 missing 824 months. The mean spectra are shown in figure 12. While the land spectrum is – as expected - 825 essentially a pure spectral plateau (with β ≈ 0.2, the value cited earlier), we see that the SST 826 spectrum is quite different displaying a clear transition between two power laws at τo ≈1 year 827 (with β ≈ 0.6, 1.8 for scales > τo and <τo respectively). Note also the rough convergence of the 828 spectra at about 100 yr scale implies that the land and ocean variability become equal and the 829 hint that there is a low frequency rise in the land spectrum for periods> ≈ 30 yrs. Since above 830 we predicted τo as about 1 year from estimates of εo, we see that the break in the empirical 831 spectrum is very close to that predicted, although compared to the land temperature spectral 832 plateau (representing more closely the free atmosphere), the low frequency ocean plateau β is a 833 little larger. In fig. 17, we will see that these conclusions are supported by direct evidence of 834 planetary scale cascades with temporal outer scales ≈ 1 yr as predicted. 835

The fact that both oceanic and atmospheric temperatures have high frequency β’s with 836 the turbulent value ≈ 1.8 support this interpretation, it also allows us to directly estimate the ratio 837

3/15/11 31

of atmospheric to oceanic energy fluxes εw/εo. This can be estimated from the left-right 838 separation of the parallel ω-1.8 lines which correspond to a factor of ≈30 in critical time scales i.e. 839 τo/τw ≈30. If we assume that the spatial outer scale for the atmosphere and oceans is the same (Lo 840 = Lw), then we can infer that εw/εo = (τo/τw )3 ≈ 3x104. Using the atmospheric value εw ≈10-3 m2s-841 3 we find εo ≈5x10-8 m2s-3 which is close to the estimates discussed above. If this interpretation is 842 correct, then, since ocean eddies and gyres obey roughly the same turbulent phenomenology as 843 the atmosphere, then the time scales <τo indeed correspond to ocean “weather”, with the 844 implication that the limits to forecasting the ocean are ≈τo ≈1 yr. 845

846 Fig. 12: This figure superposes the ocean and atmospheric plateaus showing their great 847 similarity. 848 Left: A comparison of the monthly SST spectrum (bottom, blue) and monthly atmospheric 849 temperatures over land (top, purple) for monthly temperature series from 1911-2010 on a 5ox5o 850 grid; the NOAA NCDC data). Only those near complete series (missing less than 20 months out 851 of 1200) were considered; 465 for the SST, 319 for the land series; the missing data were filled 852 using interpolation. The reference slopes correspond to β = 0.2 (top), 0.6 bottom left and 1.8, 853 bottom right. A transition at 1 year corresponds to a mean ocean εο ≈ 1x10-8m2s-3. 854 Right: The average of 5 spectra from a sections 6 years long of a thirty year series from daily 855 temperatures at a station in France (black, taken from (Lovejoy and Schertzer, 1986)). The red 856 reference line has a slope 1.8 (there is also a faint slope 0 – flat - reference line). The relative up-857 down placement of this daily spectrum with the monthly spectra (corresponding to a constant 858

3/15/11 32

factor) was determined by aligning the atmospheric spectral plateaus (i.e. the black and purple 859 spectra). 860 861

5. Cascades: generalizations of the classical turbulence laws for 862

intermittency, multifractality 863

5.1 Intermittency and Cascades 864

Up until now, we have deliberately avoided discussing the statistics of the turbulent 865 fluxes; we have only considered the generalization of the classical turbulence laws for 866 anisotropy, but anisotropy was discussed at the level of second order statistics (power spectra). 867 However, for realism we need another generalization of the classical laws to account for the 868 intermittency or “spottiness” of turbulence (Batchelor and Townsend, 1949). Although at first – 869 and even continuing today - cascades are often used in little more than a vague “conceptual” 870 invocation, quantitative explicit cascade models were developed starting with (Novikov and 871 Stewart, 1964), and by the 1980’s they were known to be the generic multifractal process. 872

There is now a large literature on cascades and multifractals, some of it was reviewed in 873 (Schertzer et al., 1997), (Lovejoy and Schertzer, 2010b), (Lovejoy and Schertzer, 2011). 874 The basic cascade idea is that starting at some external scale Leff, nonlinear interactions 875 iteratively break large structures into smaller ones with the “parents” multiplicatively modulating 876 the “daughter” fluxes. For our present purposes, we need only consider a few cascade properties, 877 notably that the general statistics of the cascades at resolution λ’ can be specified by their 878 statistical moments via the simple multifractal cascade equation: 879

! "#

q= "# K q( )

; "# = Leff / L (15) 880

where λ’ is the ratio of the “effective outer scale” Leff where the cascade begins and the 881 resolution L of the turbulent flux ϕ and K(q) is a convex function which specifies the statistics at 882 all scales. In this chapter, we will be interested in empirical analyses in which the outer scale is 883 not known a priori but is itself a significant empirically determined quantity. We will instead 884 use the symbol λ as the ratio of a convenient reference scale to the resolution; see below. 885

In general, cascades only have the weak convexity K’’>0 and scale by scale conservation 886 (<ϕλ> = constant; K(1) =0) constraints and are therefore impractical (otherwise we must deal 887 with a nearly arbitrary unknown function (K(q)), the equivalent of an infinite number of 888 parameters. Fortunately, due to a kind of multiplicative central limit theorem and under still 889 rather general circumstances, (Schertzer and Lovejoy, 1987) and (Schertzer and Lovejoy, 1997) 890 showed that cascades were expected to belong to the basin of attraction of “universal 891 multifractals” in which only two parameters C1, α are necessary: 892

K q( ) = C1q!" q( ) / ! "1( )

(16) 893 894

3/15/11 33

The parameter 0< C1< d (dimension of space) is the codimension of the mean, it characterizes 895 the sparseness of the set of points giving the dominant contribution to the mean, 0≤ α ≤ 2 is the 896 “multifractal index” which is the Levy index of the generator of the process; α = 0 corresponds 897 to a monofractal process (“β model”) and α = 2 to a Gaussian generator (Γ = logε). 898

5.2 Estimating fluxes from the fluctuations 899

In order to test the general predictions of multiplicative cascades (eq. 15), we must analyze 900 the data without relying on any specific theories of turbulence; we must use an approach that 901 does not require a priori assumptions about the physical nature of the relevant fluxes; nor of their 902 scale symmetries (isotropic or otherwise). If atmospheric dynamics are controlled by scale 903 invariant turbulent cascades of various (scale by scale) conserved fluxes ϕ then in a scaling 904 regime, the fluctuations ΔI(Δx) in an observable I (e.g. wind, temperature or radiance) over a 905 distance Δx are related to the turbulent fluxes by a relation of the form: 906

ΔI(Δx) ≈ ϕΔxH (17) 907

(eq. 17 is a generalization of the Kolmogorov law for velocity fluctuations (the latter has H = 1/3 908 and ϕ = εη, η = 1/3 where ε is the energy flux to smaller scales). Without knowing η nor H - nor 909 even the physical nature of the flux - we can use this to estimate the normalized 910 (nondimensional) flux ϕ’ at the smallest resolution (Δx = l) of our data: 911

ϕ’ = ϕ/<ϕ> = ΔI/<ΔI> (18) 912

where “<>” indicates statistical averaging. Note that if the fluxes are realizations of pure 913 multiplicative cascades then the normalized η powers !"

/ !" are also pure multiplicative 914

cascades, so that ϕ’ = ϕ/<ϕ> is a normalized cascade (<ϕ> is the ensemble mean large scale flux, 915 i.e. the climatological value, it is independent of scale, hence there is no need for a subscript). 916 The fluctuation, ΔI(Δx) can be estimated in various ways; in 1-D a convenient method (which 917 works for the common situation where 0 ≤ H ≤ 1) is to use absolute differences: 918 !I l( ) = I x + l( ) " I x( ) where l is the smallest reliable resolution and where x is a horizontal 919 coordinate, (this is sometimes called “the poor man’s wavelet”; other wavelets could be used. In 920 2-D, convenient definitions of fluctuations are the (finite difference) Laplacian (estimated as the 921 difference between the value at a grid point and the average of its neighbours), or the modulus of 922 a finite difference estimate of the gradient vector. The resulting high resolution flux estimates 923 can then be degraded (by averaging) to a lower resolution L>l. 924

Following eq. 15, the basic prediction of multiplicative cascades applied to a turbulent 925 flux is that the normalized moments: 926

Mq= !"#

q= "#

q/ "

1

q (19) 927

obey the generic multiscaling relation: 928

Mq = !" K q( )=

""eff

#

$%

&

'(

K q( )

; !" = " / "eff ; " = Le / L; "eff = Le / Leff (20) 929

where “<.>” indicates statistical (ensemble) averaging and Leff is the effective outer scale of the 930 cascade. λ is a convenient scale ratio based on the largest great circle distance on the earth: Le = 931

3/15/11 34

20,000 km and the scale ratio λ/λeff is the overall ratio from the scale where the cascade started to 932 the resolution scale L, it is determined empirically, although from the foregoing discussion we 933 expect Leff ≈ Le so that we expect λeff ≈ 1 corresponding to planetary scale cascades. Since even 934 at planetary scales each field nonlinearly interacts with the other fields, it is possible (and we 935 often find) that Leff>Le. 936

Let us now consider estimating the flux. There are two basic cases to consider. The first, 937 is widely applicable to empirical data which are nearly always sampled at scales much larger 938 than the dissipation scales, it is the one described above based on the scaling range formula, eq. 939 17. If instead we have dissipation range data (for example if we estimate fluxes from the outputs 940 of numerical models at the model dissipation scale), then the basic approach still works, but the 941 flux is generally not the same: for example in the case of the velocity field it is ε1/2 (rather than 942 ε1/3); see (Stolle et al., 2009). 943

5.3 An overview of spatial cascades 944

The spatial ECMWF cascade analyses are presented in fig. 13; for each variable we start 945 with the finite difference absolute Laplacian flux estimate which was then degraded by spatial 946 averaging in the corresponding direction and then statistical averaged over the other directions 947 (space and/or time). In the figures, one can clearly see the basic cascade structure of lines 948 converging to the external scales (this is predicted by eq. 15); note in particular that the external 949 cascade scales are systematically comparable to the largest great circle distance (20000 km), and 950 that the scaling is well respected at all but the largest scales (i.e. for log10λ> ≈ 0.6 i.e. for scales 951 <≈ 5000 km: see below for error estimates). Here and below, references to “cascade structures” 952 are simply a convenient short-hand to indicate the converging straight lines predicted for the log 953 of the moments versus log of the scale for multiplicative cascades – it does not refer to real space 954 fluid structures. The moments are only shown up to order q = 2 since for large enough q they 955 become dominated by the largest value present in the data sample so that the results spuriously 956 depend on the sample size (K(q) becomes spuriously linear; this is a “multifractal phase 957 transition” see (Schertzer and Lovejoy, 1994)). It was found that in this data, the transition 958 always occurred for q somewhat greater than 2, so that the moments shown here are well 959 estimated from the data. 960

3/15/11 35

961 Fig. 13: The analysis of the 700 mb fields at 0Z for 2006 between latitudes ±45o. The fluxes 962 were estimated using finite difference Laplacians. The curves are the moments q = 0, 0.1, 0.2, 963 …, 1.9, 2 (top). λ = 1 corresponds the size of the earth, 20,000 km. 964 965

Very similar results were found for the forecast products of the Canadian Global 966 Environment Model (GEM) at both t = 0 and t = 144, the National Weather Service Global 967 Forecasting System model (GFS), and the 20CR reanalysis (see the review in (Lovejoy and 968 Schertzer, 2011)). We find for example that the deviations are of the order ±0.3% for the 969 reanalyses, ±0.3% for GEM and ±0.5% for GFS the 20CR reanalysis is nearly the same). These 970 small deviations allow us to conclude that the analyses and models accurately have a cascade 971 structure. Overall, from the table we can also see that the K(q) “shape parameter” - the difficult 972 to estimate multifractal index α - is roughly constant at α = 1.8±0.1. 973

3/15/11 36

974 Fig. 14: Temperature (top left), relative humidity (top right), log potential temperature (lower left 975 right). From (Lovejoy et al., 2010). 976

977 Turning to in situ aircraft measurements (using the same 24 legs discussed in section 2.2) 978

we refer the reader to fig. 14 which shows the flux analysis results for the temperature and 979 humidity, potential temperature. (Lovejoy et al., 2010) analyzes this in detail including the 980 wind field which is more complex due to its sensitivity to aircraft excursions in the vertical and 981 concluding that as far as estimating horizontal scaling parameters is concerned, that the range 4 - 982 40 km is optimal (between the dashed lines in the figures): at smaller scales the trajectory is too 983 intermittent while at the longer scales, one obtains isobaric rather than isoheight statistics. We 984 nevertheless see fairly convincing cascade structure for the temperature, humidity and potential 985 temperature. Once again, the outer scales are of the order of the size of the earth although the 986 Leff for the wind is somewhat larger – the variability being presumably increased by the 987 variability of the altitude (which – due to the aircraft response to turbulence - is also cascade-like 988 with outer scale of 30 – 50 km see (Lovejoy et al., 2010)). 989

By using satellite radiance data (whose spectra were analyzed in fig. 1), (Lovejoy et al., 990 2009a) showed that the corresponding energy forcings and sinks (i.e. the short and long wave 991 radiances) are also scaling with corresponding cascade structures – including estimates of their 992 cascade parameters and outer scales. This is in accord with the spectral transfers analyzed in 993 (Lovejoy and Schertzer, 2011) which did not have a clear direction. Other spatial cascades of 994 remotely sensed data used the TRMM Precipitation Radar instrument (Lovejoy et al., 2008a) as 995 well as the TRMM a passive microwave instrument (TMI; 5 wavelengths, 2 polarizations 996

3/15/11 37

(Lovejoy et al., 2009a)). These TRMM results are bolstered by those from thermal infra red 997 data from the geostationary satellite MTSAT ((Pinel, 2010)); see also (Harvey et al., 2002), 998 (Gaonac'h et al., 2003). For reviews see (Lovejoy and Schertzer, 2010b) and (Lovejoy and 999 Schertzer, 2011) and for topography cascades, relevant ot the lower atmospheric boundary 1000 condition, see (Gagnon et al., 2006). 1001

5.4 An overview of the temporal cascade structure 1002

We have already reviewed the evidence that atmospheric fields are scaling in time up to 1003 scale τw ≈ 10 days. Given the connection between space and time and evidence that the spatial 1004 cascade structure extends out to planetary scales as reviewed in the previous subsection, we 1005 should expect cascades up to time ≈ τw although for the SST and for fields heavily influenced by 1006 the ocean, this may extend further to scales up to ≈ τo≈ 1 yr. In this section we review some of 1007 the evidence. Fig. 15 shows the first of these; the analysis of the ECMWF fluxes, the temporal 1008 analogues of the spatial analyses presented in fig. 13. Fig. 16 shows the temporal analysis of the 1009 20CR reanalysis, in this case spanning 6 hours to 138 years. The parameters are summarized in 1010 table 1. In all cases, we see the same basic features: a cascade structure which is reasonably well 1011 respected up to scales of 5 -10 days with outer cascade scales typically in the range 20 – 60 days 1012 followed by a flattening at longer time scales. Note that the outer cascade scale τeff is a bit larger 1013 than τw which is the scale at which the scaling breaks down i.e. where the empirical curves 1014 diverge from the regression lines. 1015

1016

3/15/11 38

1017 Fig. 15: The trace moments of the ECMWF interim reanalyses from daily data for 2006, the 1018 same as figs. 13 but for the temporal analyses, λ = 1 corresponds to 1 year. The effective outer 1019 temporal cascade scales (τeff) are indicated with arrows. Reproduced from (Lovejoy and 1020 Schertzer, 2010a). 1021

3/15/11 39

1022 Fig. 16: The trace moments of the spatial Laplacians of Twentieth Century reanalysis 1023

products for the band 44- 46oN for the zonal wind (upper left), meridional wind (upper right), the 1024 temperature (lower left) and specific humidity (lower right). The largest scale, λ =1 corresponds 1025 to 138 yrs, the parameters of the fits are given in table 1. Notice the “bulge” in the hs moments 1026 up to scales of ≈ 1 yr, possibly a reflection of the ocean cascade. 1027

1028 What about the predictions of section 4.3 that there is an ocean cascade to scale to ≈ 1 yr? 1029

We have seen that the TRMM precipitation has an outer scale of this order but the other fields 1030 have outer scales closer to τw. In order to see evidence of ocean cascades, we need to “filter out” 1031 the atmospheric variability, we can do this using the monthly data discussed in section 4.3 (fig. 1032 12). Using the same SST and land data sets used for the spectral analysis, we can determine the 1033 cascade structures, see fig. 17. This not only shows that the spatial temperature scaling continues 1034 to near planetary scales, but also that the temporal cascades are almost identical for land and sea 1035 temperatures with accurate scaling to about 1 year and with outer cascade scales of about 3.5 1036 years i.e. fully consistent with our estimate of τo above. To put this in perspective recall that the 1037 spectra, fig. 12 showed qualitatively and quantitatively different SST and air temperature over 1038 land spectra. Since the weather variability for τ>τw has very low intermittency, the strong 1039 intermittency of these monthly averages is presumably almost entirely due to the ocean 1040 turbulence. This scaling intermittency due to turbulent ocean processes thus leads to small 1041 statistical deviations (even in the land statistics) from perfect power law scaling for periods less 1042 than a year. This was recently noted by (Lanfredi et al., 2009) who treated the deviations as a 1043 (scale bound) Markov process rather than a scaling one. 1044

3/15/11 40

1045 Fig. 17: This compares the cascade analysis of the SST (left column) and land series (right 1046 column) for space (zonal) top row (λ =1 corresponds to 20000 km), and time, bottom row (λ =1 1047 corresponds to 100 yrs) for the data discussed in fig. 14. The parameters and sampling details 1048 are given in section 3.3. It is particularly noteworthy that although the land and ocean cascade 1049 structures are nearly identical, that the corresponding spectra (fig. 19) are very different 1050 indicating that the intermittency of the land temperatures is controlled by the ocean “weather” 1051 variability. The longest “pure” land scale accessible was ≈ 165o at midlatitudes hence the 1052 smallest accessible λ ≈ 100.2 in the upper right graph. 1053

6. Generalizing turbulent laws beyond τw: the spectral plateau, 1054 Weather- Climate cascades and dimensional transitions 1055

6.1 The fractionally Integrated Flux (FIF) model 1056

We have shown that if the classical laws of turbulence are generalized to account for 1057 anisotropic scaling (eq. 6) and for intermittency (eq.15) that they are very accurately obeyed over 1058 huge range sof space and time scales and that in the time domain, they predict a scale break at 1059 time scales corresponding to the lifetimes of planetary size structures (at τw≈ 10 days). In this 1060 section, we show how to further generalize the laws so as to reproduce the variability at time 1061 scales beyond τw; indeed up to periods of the order of 10 – 100 yrs. To make this generalization, 1062 we introduce the fractionally integrated flux (FIF) model (Schertzer and Lovejoy, 1987) which is 1063 an explicit dynamic stochastic model reproducing fields satisfying eq. 6, 15, 16. 1064

Considering just the horizontal and time domains, we have argued that the velocity scale 1065 linking time and space should be precisely the typical velocity of the largest eddy V which is 1066 determined by the external spatial scale L (the size of the planet) and the driving global mean 1067 energy flux εw (itself determined by the solar radiation modulated by the clouds and dynamics; 1068 Vw = εw

1/3Le1/3 and τw = Le/Vw. In other words, Le, Vw, τw are determined from basic principles, 1069

3/15/11 41

they are not simply adjustable model parameters. Considering the scale function for a planetary 1070 scale region with small overall mean velocity (but not small rms velocity V), we argued that for 1071 scales l < L and τ < τw, it was essentially isotropic (Geff = the identity, trivial anisotropy only 1072 given by eq. 12). 1073

While we have spent effort testing the prediction of the basic model for scales l < Le and τ 1074 < τw, we have not examined the behaviour of the model for scales τ > τw. Can the same model 1075 account for the weather – climate transition, and to what extent can it account for the climate 1076 regime (e.g. the spectral plateau)? In other words, what are the limits of the model, at what 1077 scales does it finally break down? 1078

As a step toward answering this question, we turn to explicit space-time stochastic cascade 1079 models. We first recall the basic features of (continuous in scale) cascade model for the 1080 turbulent flux ε. First, since it is assumed to be a multiplicative process, it can be expressed in 1081 terms of the exponential of an additive generator Γ: 1082

! r,t( ) = e" r ,t( ) (21) 1083 where Γ is the (dimensionless) generator and we have nondimensionalized ε by its ensemble 1084 average. If we assume that the basic statistics are translationally invariant in space-time 1085 (statistically homogeneous, statistically stationary), then Γ is given by a convolution between a 1086 basic noise γα(r,t) (independent, identically distributed random variables), and g(r,t) is a Green’s 1087 function (a deterministic weighting function that correlates them over (potentially) large space-1088 time distances): 1089

! r,t( ) = " # r,t( )$ g r,t( ) (22) 1090 For the stable and attractive processes leading to universal multifractals, γα(r,t) is taken as 1091

a unit (and extremal) Levy noise, index α, i.e. whose second characteristic function is 1092 log e

! " q = q" / " #1( ) . In addition for universal multifractals, g must have a particular form: 1093

g r,t( ) = NDC1

1/!" t( ) r,t( )!"

#$#D /!

(23) 1094 with the singularity cutoff at the inner, dissipation scale and D the dimension of space-time (= 1095 the trace of the scale generator G for isotropic space-time), ND is a normalization constant, C1 the 1096 intermittency parameter of the mean intermittency, α is the Levy index of the (extremal) 1097 uncorrelated space-time unit amplitude Levy noise γα(r,t) (see (Schertzer and Lovejoy, 1987) for 1098 the basic model, (Marsan et al., 1996) for the extension to causal space-time processes, (Lovejoy 1099 et al., 2008b) for the extension to turbulence driven waves, and (Lovejoy and Schertzer, 2009) 1100 for a technical treatment of numerical issues. Causality has been taken into account with the use 1101 of a Heaviside function Θ(t) (=0 for t< 0, = 1 for t > 0), (Marsan et al., 1996). Physically, the 1102 noise γα represnts the innovations and g the interaction strength. 1103

The observable v (e.g. a horizontal wind component nondimensionalized by the outer 1104 scale Le) whose statistics obey eqs. 13 can be obtained from the flux by taking: 1105

v r,t( ) = ! r,t( )

1/3" # t( ) r,t( )!

"#$( )

$ D$H( ) (24) 1106

This is the “Fractionally Integrated Flux” (FIF) model, (Schertzer and Lovejoy, 1987). 1107

3/15/11 42

6.2 From the weather to the climate: a dimensional transition to a new 1108

scaling regime characterized by H<0 1109

We now consider the consequences of assuming that the FIF model holds for scales τ>> 1110 τw. In order to understand the basic features that the model predicts for the weather, the 1111 transition and the climate we can restrict our attention to a D = 3, (x, y, t) section of the full (x, y, 1112 z, t) model and ignore – for the moment - the complications associated with ocean augmented 1113 intermittency which are relevant over the regime τw<Δt< τo. If we rewrite the equation for the 1114 cascade generator (eq. 22) nondimensionalizing r =(x,y) with Le and t with τw, then we obtain for 1115 the generator ! x,t( ) = log" x,t( ) : 1116

! r,t( ) = " # r $ %r ,t $ %t( )g %r , %t( )d %r d %tS&

'&w

$1

1

' + " # r $ %r ,t $ %t( )g %r , %t( )d %r d %tS&

'1

&c

' (25) 1117

!w= L

w/ L

i= "

w/ "

i is the total range of meteorological scales (Li, τi are the inner, dissipation 1118

space and time scales) and !c= "

c/ "

w is ratio of the overall outer time scale of the climate 1119 process τc to the outer time scale of the weather process; see fig. 18 for a schematic showing the 1120 ranges of integration in eq. 25. Eq. 25 is a convolution between the subgenerator noise γ which 1121 represents the “innovations” and the power law kernel g represents the interaction strength 1122 between scales physically and temporally separated by the space-time interval (r’,t’). For Λc= 1123 τc>>τw >>1 we therefore have approximately: 1124

! x,t( ) " !w x,t( ) + !c t( )

!w x,t( ) = # $ x % &x ,t % &t( )g &x , &t( )d &x d &t'w

%1

1

('w

%1

1

(

!c t( ) = # $ t % &t( )g 0, &t( )d &t1

'c

(

(26) 1125

where ! " t # $t( ) is a spatially integrated Lévy noise. The region of integration in the weather 1126 integral SΛ can be approximated by the half unit circle in (x,y,t) space with the half circle around 1127 the origin of radius Λw

-1 removed (half due to the Heaviside function, in fig. 18 only the positive 1128

quadrant is shown for clarity). The approximation in eq. 26 consists in assuming for t>> r that 1129 g(r,t) ! g(0,t) so that for long enough time lags the spatial lags are unimportant. Γw(r,t) is a 3 D 1130 (space-time) integral corresponding to the contribution to the variability from the weather regime 1131

(Λw-1<t < 1, !w

"1< r < 1), and the second Γc(r,t) is a 1-D (purely) temporal contribution due to 1132

the weather-climate regime. This drastic change of behaviour due to the change of space –time 1133 dimension over which the basic noise driving the system acts is a kind of “dimensional 1134 transition” between weather and climate processes. Figure 18 gives a schematic indicating that 1135 at small scales, the interactions occur over all spatial and temporal intervals (the interaction 1136 region is a space-time volume), whereas for long times, the interaction region is pencil-like, it is 1137 essentially 1-D. Physically this is a transition from the high frequency regime where both spatial 1138

3/15/11 43

and temporal interactions are important, to a lower frequency regime where the dynamics are 1139 dominated by temporal interactions. In the former case, this means between neighbouring 1140 structures of all sizes and at their various stages of development whereas in the latter case, only 1141 between very large structures at various stages in their development. 1142 1143

1144 Fig. 18: Schematic diagram showing the regions of integration in eq. 1 and the idea of 1145 “dimensional transition” when the region becomes pencil –like (1-D, large scales) rather than 1146 volume like (3-D, small scales), this is the dimensional transition. For clairity only the positive 1147 (x,y) quadrant is shown. 1148 1149 In this simplest model it is this separation into independent additive weather and climate 1150 generators with correlated noises integrated over spaces of different effective dimensions which 1151 is responsible for the statistical difference between weather and weather – climate plateau. At 1152 the level of the fluxes it means that the weather - climate process multiplicatively modulates the 1153 weather process at the larger time scales: 1154

!"w,"

c

r,t( ) # e$w r ,t( )+$wc r ,t( )= !

"w

r,t( )!"c

t( ) (27) 1155

with !"w

x,t( ) , having the high frequency variability, !"c

t( ) the low frequency. The generic 1156 result is a “dimensional transition” in the form of a fairly realistic spectral plateau. Physically it 1157 means that whereas at small scales, there are significant dynamical interactions in space-time, at 1158 long time scales, the spatial interactions become unimportant. 1159

Let us assume that the Green’s function g is the same as discussed in section 4 for 1160 isotropic space-time so that for D =d+1 (spatial dimensions and time): 1161

3/15/11 44

g r,t( ) = ! t( ) r,t( )!"

#$"D /#

= ! t( ) r2+ t

2( )" d+1( )/ 2#( )

(28) 1162

(i.e. taking B as the identity in eq. 11). In d spatial dimensions, from eq. 23, we see that the 1163 weather and weather-climate fluxes and generators are: 1164

!wt( ) = e"w t( )

; "w=

C1

Nd+1

#

$%&

'(

1/)* ) r + r,,t + ,t( )d ,r d ,t

,r 2+ ,t 2( )

d+1( )/ 2)( )S-

.-w

+1

1

.

!wct( ) = e"wc t( )

; "wc=

C1

Nd+1

#

$%&

'(

1/)* ) t + ,t( )d ,t

,t d+1( )/)1

-c

. (29) 1165

The exponent in the integral for Γw has been chosen so that εw is be a true cascade process with 1166 !w,"

q= "

K q( )

and K(q) obeying eq. 16, however, the same choice in the equation for Γwc falls off 1167 more rapidly than for a d = 0 (pure temporal multifractal) cascade process (which requires t-1/α), 1168 leading to low intermittency. The “plateau cascade” is called the “weather-climate” process 1169 because while the low frequency regime is new, it is ultimately predicted by the high frequency 1170 weather model; it is not a fully new climate regime. 1171

The actual statistical behaviour of this regime is actually quite complex to analyze and 1172 has some surprising properties which are investigated in detail in (Lovejoy and Schertzer, 1173 2011). The main characteristics are a) that although the bare process is still log-Levy, the weak 1174 correlations apparently lead to central limit convergence to Gaussians for the dressed statistics b) 1175

at large temporal lags Δt the autocorrelations !wct( )!

wct " #t( ) ultimately decay as Δt-1, 1176

although very large ranges of scale are necessary to observe it, c) since the spectrum is the 1177 Fourier transform of the autocorrelation and the transform of a pure Δt-1 function has a low 1178 frequency divergence, the actual spectrum of a finite range spectral plateau depends on the 1179 overall range of scales Λc, d) over surprisingly wide ranges (factors of 100 – 1000 in frequency 1180 for realistic values of Λc in the range 210 - 216), one finds “pseudo-scaling” with nearly constant 1181 spectral exponents βwc which are typically in the range 0.2 – 0.4 for realistic values of Λc, e) βwc 1182 is independent of C1 and only weakly dependent on α, f) the final fractional integration (eq. 24) 1183 for the observable has virtually no effect for τ>>τw (the kernel has a strong high frequency 1184 divergence) so that the statistics are independent of H. 1185

The upshot of this is that we expect an overall scaling with βwc whose value largely 1186 depends on the overall range of the plateau regime (Λc), pretty much independently of the values 1187 of α, C1 and H. (Lovejoy and Schertzer, 2011) gives some of the details; for example, for fits 1188 over a range a factor 128 in scale, we obtain βwc = 0.40, 0.33, 0.29, 0.23 with outer scales τc ≈ 1189 30, 110, 450, 1800 yrs (with α = 1.8, τw = 10 dys, i.e. corresponding to Λc = 210, 212, 214, 216). 1190 The (rough) empirical range of βwc ≈ 0.2 – 0.4 is thus compatible with τc ≈ > 30 yrs. In 1191 summary, we therefore find for the overall FIF model: 1192

3/15/11 45

E k( ) ! k"#w ;

E $( ) ! $"#w ;

E $( ) ! $"#wc ;

k > Lw

"1

$ > %w

"1

%c

"1< $ < %

w

"1

(30) 1193

where τc is the long external scale where the plateau ends (see discussion below) and the weather 1194 and weather - climate spectral exponents are: 1195

!w= 1+ 2H " K 2( )

0.2 < !wc< 0.4

(31) 1196

As expected, the high frequency weather exponent is the usual one (with the usual structure 1197 function exponent ξ(q) = qH-K(q)), but the low frequency weather - climate exponent βwc is new. 1198 Note in particular that it is independent of H and that βwc >0. As we just argued, in the weather-1199 climate regime, the intermittency rapidly disappears as we “dress” the process by averaging over 1200 scales >τw so that the weather-climate plateau is nearly monofractal and we have an effective 1201 climate exponent Hwc: 1202

Hwc! " 1" #

wc( ) / 2 (32) 1203

so that (since βwc<1), Hwc<0 and the corresponding (generalized) structure function exponent at 1204 least approximately satisfies the monofractal linear relation: 1205

!wcq( ) " qHwc (33) 1206

Using 0.2 < βwc < 0.4 corresponding to -0.4 < Hwc < -0.3, this result already explains the 1207 preponderance of spectral plateau β’s around that value already noted in many analyses 1208 presented in section 3. However, as we saw in fig. 12, the ocean plateau has a somewhat higher 1209 βoc ≈ 0.6 which implies Hoc ≈ -0.2; in (Lovejoy and Schertzer, 2011) we use a simple coupled 1210 ocean - atmosphere model to show how this could arise as a consequence of double (atmosphere 1211 and ocean) dimensional transitions. 1212

6.3 Testing the FIF weather/climate model… or how to determine 1213

decadal scale climate variability from 1 Hz aircraft data 1214

In order to test the realism of the FIF model in reproducing the weather-climate plateau, 1215 we made a detailed comparison of temperature data and numerical simulations of the FIF 1216 weather model. The data were taken at 75o N (taken from the 20CR reanalysis, from 1871-1217 2008). We averaged from 6 hourly to daily resolutions which resulted in a series 50404 days 1218 long for each longitudinal pixel. The simulation was of a simple (x,t) cascade (see fig. 19) with 1219 the parameters: α =1.8, C1=0.1, Hw =0.5, εw ≈ 5x10-4 m2/s3 (hence τw =10 days) observed by the 1220 NOAA Gulfstream 4 aircraft near 200 mb altitude; see table 1. The simulation was scaled so that 1221 its standard deviation coincided with that of the data. The latitude 75o N was chosen both 1222

3/15/11 46

because it is largely dominated by land or ice covered ocean so that intermittency effects due to 1223 the ocean circulation are not so strong, and because later we discuss and compare it with 1224 Greenland paleotemperatures which are at about this latitude. 1225

Except for the standard deviation and mean of the temperature at 75o N there was no 1226 attempt whatsoever to “fit” the simulation to the reanalysis data so that we do not expect a 1227 perfect data/simulation match. The object was to see how close the above “toy model” of a 1228 dimensional transition can account for the atmospheric variability over large ranges of time 1229 scales. In order to remove extraneous issues of sample size and series length, a 217 x 24 1230 simulation was made and the first 50404 spatial segments were taken (this is a bit less than half 1231 the length and so avoids artificial correlations/ tendencies due to the periodicity of the 1232 simulations). 1233 1234

1235 Fig. 19: Top: 700 mb temperature data from the 20CR reanalysis (1871-2008) for 2ox2o pixels at 1236 75N; the data was detrended and averaged over 16 days (1871 at left, 2008 at right); 16 pixels 1237 spaced at 10o in longitude were used so as to exactly match the simulation. Below, we show a 1238 time series of a single pixel of an (x,t) simulation 16x50404 pixels and then averaged over 16 1239 simulated days; the effective scales of the simulation were 1 day in time and Lw/16≈ 1200 km in 1240 space, the parameters were α = 1.8, C1 = 0.1, Hw = 0.5 (close to the temperature as measured by 1241

3/15/11 47

aircraft). For graphical purposes, in addition to the standard deviation (= ±4.05 K), the 1242 simulation mean was adjusted to be the same as the data. 1243

1244 Fig. 20: A comparison of the spectrum of the same data as in fig. 20 (bottom, blue points), with 1245 the simulation (top, red points); both spectra were averaged over logarithmic bins, 10 per order 1246 of magnitude. The reference lines have the theoretical slopes βw, βwc indicated in eq. 31. 1247 1248

The comparison of the simulation and empirical spectra is shown in fig. 20. For clarity, 1249 we have averaged the spectrum over logarithmically spaced bins, ten per order of magnitude. 1250 Over the high frequency weather regime, the model and data agree quite well. This is not 1251 surprising since the aircraft data that were used to determine the parameters were at these smaller 1252 scales (recall that the overall standard deviations are the same; this determines the relative 1253 vertical placement of the curves in fig. 20). However what is not at all trivial is that the low 1254 frequency part of the spectrum - including the mean spectral exponent β ≈ 0.2 - is also quite well 1255 reproduced; presumably the agreement would be better if the critical external scale was given a 1256 small adjustment. Certainly, ensemble averaging over many realizations of the FIF model give a 1257 fairly accurate slope β = 0.2 as indicated by the reference line. The high frequency aircraft 1258 parameters thus give a remarkably good model of the temperature spectra at 75oN out to 1259 decadal/centennial scales. In section 4.2 we saw that at somewhat larger scales, the climate 1260 regime proper emerges and the model is no longer accurate. 1261

We should however mention that for more realism we should couple the ocean and 1262 atmosphere, simple models for this not only reproduce the cascade structure of monthly data (fig. 1263 17), but also the otherwise anomalous exponent β≈ 0.6, further discussion of this model 1264 (including intermittency). For further model-data intercomparisons, see (Lovejoy and 1265 Schertzer, 2011). 1266

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7. Conclusions 1267

Just the laws of continuum mechanics emerge from those of statistical mechanics when 1268 large enough numbers of particles are present, so do the laws of turbulence emerge at high 1269 enough Reynold’s numbers, strong enough nonlinearity. However, the classical turbulence laws 1270 were constrained by strong assumptions of homogeneity and isotropy and could not cover wide 1271 scale ranges in the atmosphere. By generalizing them to take into account anisotropy (especially 1272 vertical stratification) and intermittency, their range of applicability is vastly increased. In the 1273 last five years - thanks in part to the ready availability of huge global scale data sets of all kinds - 1274 it has been possible to verify that these generalized emergent laws accurately hold up to 1275 planetary scales, for a recent overview see (Lovejoy and Schertzer, 2010b). 1276

These “weather regime” laws show that the horizontal variability is fundamentally 1277 controlled by solar forcing via the energy flux. First principles calculations show that this 1278 accurately accounts for the large scale winds and predicts a drastic “dimensional transition” at τw 1279 ≈10 days, the typical lifetime of planetary scale structures. Beyond this time scale, spatial 1280 interactions are rapidly quenched so that the longer scales are driven by essentially temporal 1281 degrees of freedom and the spectra of atmospheric fields display a shallow “spectral plateau”. 1282 We show that by making a third generalization of the classical laws that the behaviour in this 1283 “weather-climate” regime can be accurately predicted. The main complication is that - due to 1284 similar effects from ocean turbulence whose corresponding outer time scale τo ≈ 1 year – there is 1285 enhanced intermittency up to that scale, with a slightly steeper “ocean plateau” beyond. 1286 Depending on the field and location, the plateau continues up to scales of ≈ 10 – 100 years 1287 beyond which the climate regime proper begins. 1288

In this overview, we first review the basic evidence for wide range space-time scaling, 1289 first at weather scales, then at longer weather-climate (plateau) and climate scales using paleo-1290 temperatures to quantify the variability up to scales of over 100 kyrs. We then review the basic 1291 theory for both anisotropic scaling (Generalized Scale Invariance) and for intermittency 1292 (multifractal cascades). Finally, we outline the third generalization to time scales >τw using the 1293 Fractionally Integrated Flux model. We give an example comparing numerical simulations with 1294 reanalysis data showing that the model works quite well. The overall picture is of three basic 1295 scaling ranges operating from milliseconds to ≈ 100 kyrs, although we saw that the intermediate 1296 weather-climate regime is really an extension of the higher frequency atmospheric and “oceanic” 1297 weather. Rather than the climate being no more than long term weather, it thus emerges 1298 governed by new (scaling) laws. Weather and climate can thus be objectively distinguished by 1299 their different types of scaling variability, we can objectively answer the question “what is the 1300 climate”? 1301

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A. Lazarev, D. Schertzer, S. Lovejoy and Y. Chigirinskaya, Unified multifractal atmospheric 1367 dynamics tested in the tropics: part II, vertical scaling and Generalized Scale Invariance, 1368 Nonlinear Processes in Geophysics 1(1994), pp. 115-123. 1369

P.Y. Le Traon, P. Klein, B.L. Hua and G. Dibarboure, Do Altimeter Wavenumber Spectra Agree 1370 With The Interior Or Surface Quasigeostrophic Theory? , J. of Phys. Oceanography 1371 38(2008), pp. 1137-1142. 1372

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R.-C. Lien and E.A. D’Asaro, Measurement Of Turbulent Kinetic Energy Dissipation Rate With 1375 A Lagrangian Float, J. Atmos. Oceanic Tech. 23(2006), pp. 964-976. 1376

S. Lovejoy et al., Universal Multfractals and Ocean patchiness Phytoplankton, physical fields 1377 and coastal heterogeneity, J. Plankton Res. 23(2000), pp. 117-141. 1378

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S. Lovejoy and D. Schertzer, On the simulation of continuous in scale universal multifractals, 1381 part I: spatially continuous processes, Computers and Geoscience(2009). 1382

S. Lovejoy and D. Schertzer, Space-time cascades and the scaling of ECMWF reanalyses: fluxes 1383 and fields, J. Geophys. Res.(2010a), p. (in press). 1384

S. Lovejoy and D. Schertzer, Towards a new synthesis for atmospheric dynamics: space-time 1385 cascades, Atmos. Res. 96(2010b), pp. pp. 1-52, doi:10.1016/j.atmosres.2010.1001.1004. 1386

S. Lovejoy and D. Schertzer, Multifractal Cascades and the emergence of Atmospheric 1387 Dynamics, Cambridge University Press, Cambridge (2011) 400 pp. 1388

S. Lovejoy, D. Schertzer and V. Allaire, The remarkable wide range scaling of TRMM 1389 precipitation, Atmos. Res. 10.1016/j.atmosres.2008.02.016(2008a). 1390

S. Lovejoy et al., Atmospheric complexity or scale by scale simplicity? , Geophys. Resear. Lett. 1391 36(2009a), pp. L01801, doi:01810.01029/02008GL035863. 1392

S. Lovejoy, D. Schertzer, M. Lilley, K.B. Strawbridge and A. Radkevitch, Scaling turbulent 1393 atmospheric stratification, Part I: turbulence and waves, Quart. J. Roy. Meteor. 1394 Soc.(2008b), p. DOI: 10.1002/qj.1201. 1395

S. Lovejoy , A. Tarquis, H. Gaonac'h and D. Schertzer, Single and multiscale remote sensing 1396 techniques, multifractals and MODIS derived vegetation and soil moisture, Vadose Zone 1397 J. submitted 2007(2007). 1398

S. Lovejoy, A.F. Tuck, S.J. Hovde and D. Schertzer, Is isotropic turbulence relevant in the 1399 atmosphere?, Geophys. Res. Lett. L14802, doi:10.1029/2007GL029359.(2007). 1400

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S. Lovejoy, A.F. Tuck and D. Schertzer, The Horizontal cascade structure of atmospheric fields 1401 determined from aircraft data, J. Geophys. Res. 115(2010), p. D13105. 1402

S. Lovejoy, A.F. Tuck, D. Schertzer and S.J. Hovde, Reinterpreting aircraft measurements in 1403 anisotropic scaling turbulence, Atmos. Chem. and Phys. 9(2009b), pp. 1-19. 1404

S. Lovejoy, B. Watson, Y. Grosdidier and D. Schertzer, Scattering in Thick Multifractal Clouds, 1405 Part II: Multiple Scattering, 388(2009c), pp. 3711-3727. 1406

B.B. Mandelbrot, How long is the coastline of Britain? Statistical self-similarity and fractional 1407 dimension, Science 155(1967), pp. 636-638. 1408

D. Marsan, D. Schertzer and S. Lovejoy, Causal space-time multifractal processes: predictability 1409 and forecasting of rain fields, J. Geophy. Res. 31D(1996), pp. 26,333-326,346. 1410

T. Matsuno, J.-S. Lee, M. Shimizu, S.-H. Kim and I.-C. Pang, Measurements of the turbulent 1411 energy dissipation rate and an evaluation of the dispersion process of the Changjiang 1412 Diluted Water in the East China Sea, , J. Geophys. Resear. 111(2006), pp. C11S09, 11-1413 14. 1414

W. McLeish, Spatial spectra of ocean surface temperature, J. Geophys. Res. 75(1970), pp. 6872–1415 6877. 1416

J.M. Mitchell, An overview of climatic variability and its causal mechanisms, Quaternary Res. 1417 6(1976), pp. 481-493. 1418

A.S. Monin, Weather forecasting as a problem in physics, MIT press, Boston Ma (1972). 1419 A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics, MIT press, Boston MA (1975). 1420 J.N. Moum, M.C. Gregg and R.C. Lien, Comparison of turbulent Kinetic energy dissipation rate 1421

estimates from two ocean microstructure profiler, , J. Ocean. and Atmos. Tech. 12(1995), 1422 pp. 346-366. 1423

H. Nakajima and N. Hayakawa, A Cross-Correlation Analysis of Tidal Current, Water 1424 Temperature and Salinity Records, Journal of the Oceanographical Society of Japan 1425 38(1982), pp. 52 - 56. 1426

E.A. Novikov and R. Stewart, Intermittency of turbulence and spectrum of fluctuations in 1427 energy-disspation, Izv. Akad. Nauk. SSSR. Ser. Geofiz. 3(1964), pp. 408-412. 1428

E. Palmén, The atmosphere and the Sea in Motion. In: B. Bolen, Editor, Oxford Univ, Press 1429 (1959), pp. 212-224. 1430

H.A. Panofsky, The spectrum of temperature, J. of Radio Science 4(1969), pp. 1101-1109. 1431 J. Pelletier, D., The power spectral density of atmospheric temeprature from scales of 10**-2 to 1432

10**6 yr, , EPSL 158(1998), pp. 157-164. 1433 J. Perrin, Les Atomes, NRF-Gallimard, Paris (1913). 1434 T.C. Peterson and R.S. Vose, An overview of the Global Historical Climatology Network 1435

temperature database, Bull. Amer. Meteorol. Soc. 78(1997), pp. 2837-2849. 1436 J. Pinel, The space-time cascade structure of the atmosphere. Physics, McGill, Montreal (2010), 1437

p. (in preparation). 1438 N.Z. Pinus, The energy of atmospheric macro-turbulence, Izvestiya, Atmospheric and Oceanic 1439

Physics 4(1968), p. 461. 1440 N.A. Rayner et al., Improved analyses of changes and uncertainties in marine temperature 1441

measured in situ since the mid-nineteenth century: the HadSST2 dataset. , J. Climate 1442 19(2006), pp. 446-469. 1443

L.F. Richardson, The problem of contiguity: an appendix of statistics of deadly quarrels, General 1444 Systems Yearbook 6(1961), pp. 139-187. 1445

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C. Robert, W., Turbulent energy dissipation in the Atlantic equatorial undercurrent. U. of British 1446 Columbia, Vancouver, B.C. (1976). 1447

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D. Schertzer and S. Lovejoy, Universal Multifractals do Exist!, J. Appl. Meteor. 36(1997), pp. 1464 1296-1303. 1465

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D. Schertzer, I. Tchiguirinskaia, S. Lovejoy and A.F. Tuck, Quasi-geostrophic turbulence and 1468 generalized scale invariance, a theoretical reply to Lindborg, Atmos. Chem. and Physics. 1469 Disscussion (in preparation)(2010). 1470

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T.M.e.a. Smith, Improvements to NOAA's Historical Merged Land-Ocean Surface 1478 Temperature Analysis (1880-2006), J. Climate 21(2008), pp. 2283-2293. 1479

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daily temperatures, Phys. Rev. E 62(2000), pp. 150-160. 1484 G.I. Taylor, The spectrum of turbulence, Proceedings of the Royal Society of London A 1485

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1504 1505

1506

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Figure Captions: 1506

1507

1508 Fig. 1: Spectra from ≈ 1000 orbits of the VIRS instrument (Visible Infrared Scanner) on the 1509 TRMM satellite channels 1-5 (at wavelengths of 0.630, 1.60, 3.75, 10.8, 12.0 µm from top to 1510 bottom, displaced in the vertical for clarity). The data are for the period January through March 1511 1998 and have nominal resolutions of 2.2 km. The straight regression lines have spectral 1512 exponents β = 1.35, 1.29, 1.41, 1.47, 1.49 respectively, close to the value β =1.53 corresponding 1513 to the spectrum of passive scalars (= 5/3 minus intermittency corrections, see ch.3). The units 1514 are such that k = 1 is the wavenumber corresponding to the size of the planet (20000 km)-1. 1515 Channels 1, 2 are reflected solar radiation so that only the 15600 km sections of orbits with 1516 maximum solar radiation were used. The high-wavenumber falloff is due to the finite resolution 1517 of the instruments. Adapted from (Lovejoy et al., 2008b). 1518

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1519 Fig. 2: Inter-comparisons of the spectra of different atmospheric fields from the ECMWF interim 1520 reanalysis. Top (red) is the geopotential (β = 3.35), second from the top (green) is the zonal wind 1521 (β = 2.40), 3rd from the top (cyan) is the meridional wind (β = 2.40), 4th from the top (blue) is the 1522 temperature (β = 2.40) 5th from the top (orange) is the vertical wind (β = 0.4), at the bottom, 1523 (purple) is the specific humidity (β = 1.6). All are at 700 mb and between ±45o latitude, every 1524 day in 2006 at GMT. The scale at the far left corresponds to 20000 km in the east-west direction, 1525 at the far right to 660 km. Note that for these 2-D spectra, Gaussian white noise would yield β = 1526 -1 (a positive slope = +1). 1527

1528

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1528 1529

1530 Fig. 3: Aircraft spectra of temperature (blue, bottom), humidity (red, middle), log potential 1531 temperature (gold, top), reference lines β = 2. These are averages over 24 isobaric aircraft 1532 “legs” near 200 mb taken over the Pacific Ocean during the Pacific Winter Storms 2004 1533 experiment, the resolution was 280 m (Nyquist wavenumber = (560 m)-1. Adapted from 1534 (Lovejoy et al., 2010). 1535

1536

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1536 1537

1538 Fig. 4: Wind spectra from 36 near complete 60 year series of daily data from the continental US 1539 taken from the stations lying nearest to 2ox2o degree grid points from 30o-50 o N, 105 o to 71 o W. 1540 The brown curve (lower at left) is the daily maximum wind speed and the green is daily average 1541 (normalized so that the annual peaks coincide). The spectra have been also averaged in the 1542 frequency domain; in bins logarithmically spaced, 10 per order of magnitude (except for the 1543 lowest factor of 10 where no spectral averaging was performed). The reference lines have 1544 absolute slopes β = 1 (low frequencies), β = 0.2 (the plateau), and β =1.6 (high frequencies). 1545

1546

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1546 1547

1548 Fig. 5: This shows the scaling of hourly surface temperatures from 4 stations in the northwest 1549 US, for 4 years (2005-2008) from the US Climate Reference Network. One can see that in spite 1550 of the strong diurnal cycle (and harmonics), that the basic scaling extends to about 7 days. The 1551 reference lines (with absolute slopes 0.2, 2 are theoretically motivated, see section 6. The 1552 spectra of hourly surface temperature data from 4 nearly colinear stations running north west - 1553 south east in the US (Lander, WY, Harrison NE, Whitman NE, Lincoln NE), from the US 1554 Climate Reference Network, 2005-2008. The thick line is the spectrum of the periodically 1555 detrended spectrum, averaged over logarithmically spaced bins. 1556

1557

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1557 1558

1559 Fig. 6: Mean spectrum of daily dew point temperature, Td, temperature, T and relative humidity, 1560 h for 36, 33, 7 stations respectively, (numbers vary due to missing data) from stations with long 1561 (60 year; 22200 days) records. The low frequency and the “plateau” reference lines have slopes 1562 -1 and -0.2 respectively. The spectra were averaged over 1dBω bins (i.e. 10 per order of 1563 magnitude in frequency), every 2o from 30o-50o latitude, from -105o to -71o longitude. The high 1564 frequency reference line has absolute slope β =2 close to the horizontal β value for the humidity 1565 and temperature which are each about 1.8 - 1.9 (see table 1), and the plateau value is very close 1566 to the theory value 0.2. Adapted from (Lovejoy and Schertzer, 2010b). 1567

1568

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1568 1569

1570 Fig. 7: The temporal spectra of the daily 700 mb ECMWF interim reanalysis fields that were 1571 analyzed spatially in fig. 2 (i.e. between ±45o latitude). The dashed slopes are 5/3, the frequency 1572 spectrum is shown estimated using ensemble and spectral averaging, units of ω: cycles/year. 1573 The dashed lines have slopes -5/3, the solid lines have slopes -3.35, -0.4, -2.4, -1.1 (top to 1574 bottom); they are drawn for ω > (11 days)-1. These correspond to the spatial z exponent, the 1575 spatial w exponent (which accurately fits hs), the spatial u, v, T exponent and the regression w 1576 slope respectively. The curves top (on left) to bottom are z, hs (multiplied by 10 from spatial 1577 analysis, i.e. the spectra are multiplied by 100), v, T, u and w respectively. Note that the low 1578 frequency rise is due to only two frequencies (1 yr)-1 and (6 months)-1, it is a low resolution 1579 artefact of the annual cycle and subharmonic, not a true break in the scaling (see e.g. figs. 5, 8). 1580 Reproduced from (Lovejoy and Schertzer, 2010a). 1581

1582

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1582 1583

1584 Fig. 8 a: Spectra from the 20CR reanalysis (1871-2008) at 45oN. The reference lines have 1585 correspond to βοc = 0.6, βwc =0.2, βw =2 left to right respectively. 1586

1587

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1587 1588

1589 Fig. 8 b: Spectra from the 20CR reanalysis (1871-2008) at 5oN. The reference lines correspond 1590 to βoc = 0.6, βwc =0.2, βw =2 left to right respectively. 1591

1592

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1592 1593

1594 Fig. 9: The spectrum from 20CR reanalysis (1891-2002); the energy flux estimated from the 1595 absolute Laplacian of the zonal wind at 700 mb, 42oN. The reference lines have βc = 2, βwc = 1596 0.2, βw =1. Note the daily and annual spikes and subharmonics. 1597

1598

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1598 1599

1600 Fig. 10: The spectra (averaged over logarithmically spaced bins, 10 per order of magnitude, 1601 using the same colours as previous). The units are such that ω = 1 corresponds to (129 yrs)-1; 1602 note the annual spike, (1 year)-1 is at 2.11 on the log10ω axis). 1603

1604

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1604 1605

1606 Fig. 11: A modern composite based purely on the GRIP core and 20CR reanalyses. All spectra 1607 have been averaged over logarithmically spaced bins, 10 per order of magnitude. Spectra of 1608 northern hemisphere temperatures (red; from the 20 CR reanalysis, 1871-2008), blue is the single 1609 2 ox2o) pixel spectrum at 75oN also from the same reanalysis (and at the same scale). The light 1610 green is the mean of the GRIP high for last 90 kyrs and the (lowest) frequency blue is from the 1611 GRIP 55 cm core interpolated to 200 yr resolution and going back 240 kyrs. The solid reference 1612 lines have absolute slopes βwc = 0.2, βwc = 0.6 and βc = 1.4 and βw = 2 as indicated. The dashed 1613 line has absolute slopes βwc = 0.6 which is suggested by the global scale and equatorial analyses 1614 in fig. 8. The red arrows at the bottom (and upper right) indicate the basic qualitatively different 1615 scaling regimes. Reproduced from (Lovejoy and Schertzer, 2011). 1616

1617

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1617 1618

1619

1620 Fig. 12: This figure superposes the ocean and atmospheric plateaus showing their great 1621 similarity. 1622 Left: A comparison of the monthly SST spectrum (bottom, blue) and monthly atmospheric 1623 temperatures over land (top, purple) for monthly temperature series from 1911-2010 on a 5ox5o 1624 grid; the NOAA NCDC data, see table 5 for details). Only those near complete series (missing 1625 less than 20 months out of 1200) were considered; 465 for the SST, 319 for the land series; the 1626 missing data were filled using interpolation. The reference slopes correspond to β = 0.2 (top), 1627 0.6 bottom left and 1.8, bottom right. A transition at 1 year corresponds to a mean ocean εο ≈ 1628 1x10-8m2s-3. 1629 Right: The average of 5 spectra from a sections 6 years long of a thirty year series from daily 1630 temperatures at a station in France (black, taken from (Lovejoy and Schertzer, 1986)). The red 1631 reference line has a slope 1.8 (there is also a faint slope 0 – flat - reference line). The relative up-1632 down placement of this daily spectrum with the monthly spectra (corresponding to a constant 1633 factor) was determined by aligning the atmospheric spectral plateaus (i.e. the black and purple 1634 spectra). 1635

1636

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1636 1637

1638 Fig. 13: The analysis of the 700 mb fields at 0Z for 2006 between latitudes ±45o. The fluxes 1639 were estimated using finite difference Laplacians. The curves are the moments q = 0, 0.1, 0.2, 1640 …, 1.9, 2 (top). λ = 1 corresponds the size of the earth, 20,000 km. 1641

1642

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1642 1643

1644 Fig. 14: Temperature (top left), relative humidity (top right), log potential temperature (lower left 1645 right). From (Lovejoy et al., 2010). 1646

1647

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1647

1648 Fig. 15: The trace moments of the ECMWF interim reanalyses from daily data for 2006, the 1649 same as figs. 13 but for the temporal analyses, λ = 1 corresponds to 1 year. The effective outer 1650 temporal cascade scales (τeff) are indicated with arrows. Reproduced from (Lovejoy and 1651 Schertzer, 2010a). 1652

1653

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1653 1654

1655 Fig. 16: The trace moments of the spatial Laplacians of Twentieth Century reanalysis products 1656 for the band 44- 46oN for the zonal wind (upper left), meridional wind (upper right), the 1657 temperature (lower left) and specific humidity (lower right). The largest scale, λ =1 corresponds 1658 to 138 yrs, the parameters of the fits are given in table 1. Notice the “bulge” in the hs moments 1659 up to scales of ≈ 1 yr, possibly a reflection of the ocean cascade. 1660

1661

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1661 1662

1663

1664 Fig. 17: This compares the cascade analysis of the SST (left column) and land series (right 1665 column) for space (zonal) top row (λ =1 corresponds to 20000 km), and time, bottom row (λ =1 1666 corresponds to 100 yrs) for the data discussed in fig. 14. The parameters and sampling details 1667 are given in section 3.3. It is particularly noteworthy that although the land and ocean cascade 1668 structures are nearly identical, that the corresponding spectra (fig. 19) are very different 1669 indicating that the intermittency of the land temperatures is controlled by the ocean “weather” 1670 variability. The longest “pure” land scale accessible was ≈ 165o at midlatitudes hence the 1671 smallest accessible λ ≈ 100.2 in the upper right graph. 1672

1673

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1673 1674

1675 Fig. 18: Schematic diagram showing the regions of integration in eq. 1 and the idea of 1676 “dimensional transition” when the region becomes pencil –like (1-D, large scales) rather than 1677 volume like (3-D, small scales), this is the dimensional transition. For clairity only the positive 1678 (x,y) quadrant is shown. 1679

1680

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1680

1681 Fig. 19: Top: 700 mb temperature data from the 20CR reanalysis (1871-2008) for 2ox2o pixels at 1682 75N; the data was detrended and averaged over 16 days (1871 at left, 2008 at right); 16 pixels 1683 spaced at 10o in longitude were used so as to exactly match the simulation. Below, we show a 1684 time series of a single pixel of an (x,t) simulation 16x50404 pixels and then averaged over 16 1685 simulated days; the effective scales of the simulation were 1 day in time and Lw/16≈ 1200 km in 1686 space, the parameters were α = 1.8, C1 = 0.1, Hw = 0.5 (close to the temperature as measured by 1687 aircraft). For graphical purposes, in addition to the standard deviation (= ±4.05 K), the 1688 simulation mean was adjusted to be the same as the data. 1689

1690

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1690

1691 Fig. 20: A comparison of the spectrum of the same data as in fig. 20 (bottom, blue points), with 1692 the simulation (top, red points); both spectra were averaged over logarithmic bins, 10 per order 1693 of magnitude. The reference lines have the theoretical slopes βw, βwc indicated in eq. 31. 1694 1695