Search

Temperature and mineral dust records serve as valuable palaeoclimatic indicators for studying atmospheric variability across different temporal scales. In this study, we employed Haar fluctuations to analyse global spatiotemporal atmospheric variability over the Last Glacial Cycle, capturing both high- and low-frequency information within the records, regardless of uniform or non-uniform sampling. Furthermore, we utilised Haar fluctuations to compute fluctuation correlations, thereby enhancing our understanding of palaeoclimate dynamics. Our findings reveal a latitudinal dependency in the transition from macroweather to climate regimes (τc), with polar regions experiencing shorter transitions compared to the tropics and mid-latitudes. These transitions occur at approximately 1/100th of glacial cycle length scales, suggesting a dominant forcing mechanism beyond Milankovitch cycles. Additionally, our analysis shows that polar regions have larger fluctuation amplitudes than lower latitudes as a consequence of the polar amplification effect. Furthermore, fluctuation correlations demonstrate faster synchronisation between the poles themselves compared to lower-latitude sites, achieving high correlation values within 10 kyr. Therefore, our findings suggest a consistent climate signal propagating from the poles to the Equator, representing the first empirical evidence supporting the hypothesis that the poles play a pivotal role as climate change drivers, influencing the variability in climatic transitions worldwide.

Until the 1980s, scaling notions were restricted to self-similar homogeneous special cases. I review developments over the last decades, especially in multifractals and generalized scale invariance (GSI). The former is necessary for characterizing and modelling strongly intermittent scaling processes, while the GSI formalism extends scaling to strongly anisotropic (especially stratified) systems. Both of these generalizations are necessary for atmospheric applications. The theory and some of the now burgeoning empirical evidence in its favour are reviewed. Scaling can now be understood as a very general symmetry principle. It is needed to clarify and quantify the notion of dynamical regimes. In addition to the weather and climate, there is an intermediate “macroweather regime”, and at timescales beyond the climate regime (up to Milankovitch scales), there is a macroclimate and megaclimate regime. By objectively distinguishing weather from macroweather, it answers the question “how long does weather last?”. Dealing with anisotropic scaling systems – notably atmospheric stratification – requires new (non-Euclidean) definitions of the notion of scale itself. These are needed to answer the question “how big is a cloud?”. In anisotropic scaling systems, morphologies of structures change systematically with scale even though there is no characteristic size. GSI shows that it is unwarranted to infer dynamical processes or mechanisms from morphology. Two “sticking points” preventing more widespread acceptance of the scaling paradigm are also discussed. The first is an often implicit phenomenological “scalebounded” thinking that postulates a priori the existence of new mechanisms, processes every factor of 2 or so in scale. The second obstacle is the reluctance to abandon isotropic theories of turbulence and accept that the atmosphere's scaling is anisotropic. Indeed, there currently appears to be no empirical evidence that the turbulence in any atmospheric field is isotropic. Most atmospheric scientists rely on general circulation models, and these are scaling – they inherited the symmetry from the (scaling) primitive equations upon which they are built. Therefore, the real consequence of ignoring wide-range scaling is that it blinds us to alternative scaling approaches to macroweather and climate – especially to new models for long-range forecasts and to new scaling approaches to climate projections. Such stochastic alternatives are increasingly needed, notably to reduce uncertainties in climate projections to the year 2100.

We consider the statistical properties of solutions of the stochastic fractional relaxation equation and its fractionally integrated extensions that are models for the Earth's energy balance. In these equations, the highest-order derivative term is fractional, and it models the energy storage processes that are scaling over a wide range. When driven stochastically, the system is a fractional Langevin equation (FLE) that has been considered in the context of random walks where it yields highly nonstationary behaviour. An important difference with the usual applications is that we instead consider the stationary solutions of the Weyl fractional relaxation equations whose domain is −∞ to t rather than 0 to t. An additional key difference is that, unlike the (usual) FLEs – where the highest-order term is of integer order and the fractional term represents a scaling damping – in the fractional relaxation equation, the fractional term is of the highest order. When its order is less than 1/2 (this is the main empirically relevant range), the solutions are noises (generalized functions) whose high-frequency limits are fractional Gaussian noises (fGn). In order to yield physical processes, they must be smoothed, and this is conveniently done by considering their integrals. Whereas the basic processes are (stationary) fractional relaxation noises (fRn), their integrals are (nonstationary) fractional relaxation motions (fRm) that generalize both fractional Brownian motion (fBm) as well as Ornstein–Uhlenbeck processes. Since these processes are Gaussian, their properties are determined by their second-order statistics; using Fourier and Laplace techniques, we analytically develop corresponding power series expansions for fRn and fRm and their fractionally integrated extensions needed to model energy storage processes. We show extensive analytic and numerical results on the autocorrelation functions, Haar fluctuations and spectra. We display sample realizations. Finally, we discuss the predictability of these processes which – due to long memories – is a past value problem, not an initial value problem (that is used for example in highly skillful monthly and seasonal temperature forecasts). We develop an analytic formula for the fRn forecast skills and compare it to fGn skill. The large-scale white noise and fGn limits are attained in a slow power law manner so that when the temporal resolution of the series is small compared to the relaxation time (of the order of a few years on the Earth), fRn and its extensions can mimic a long memory process with a range of exponents wider than possible with fGn or fBm. We discuss the implications for monthly, seasonal, and annual forecasts of the Earth's temperature as well as for projecting the temperature to 2050 and 2100.

We produce climate projections through the 21st century using the fractional energy balance equation (FEBE): a generalization of the standard energy balance equation (EBE). The FEBE can be derived from Budyko–Sellers models or phenomenologically through the application of the scaling symmetry to energy storage processes, easily implemented by changing the integer order of the storage (derivative) term in the EBE to a fractional value. The FEBE is defined by three parameters: a fundamental shape parameter, a timescale and an amplitude, corresponding to, respectively, the scaling exponent h, the relaxation time τ and the equilibrium climate sensitivity (ECS). Two additional parameters were needed for the forcing: an aerosol recalibration factor α to account for the large aerosol uncertainty and a volcanic intermittency correction exponent ν. A Bayesian framework based on historical temperatures and natural and anthropogenic forcing series was used for parameter estimation. Significantly, the error model was not ad hoc but rather predicted by the model itself: the internal variability response to white noise internal forcing. The 90 % credible interval (CI) of the exponent and relaxation time were h=[0.33, 0.44] (median = 0.38) and τ=[2.4, 7.0] (median = 4.7) years compared to the usual EBE h=1, and literature values of τ typically in the range 2–8 years. Aerosol forcings were too strong, requiring a decrease by an average factor α=[0.2, 1.0] (median = 0.6); the volcanic intermittency correction exponent was ν=[0.15, 0.41] (median = 0.28) compared to standard values α=ν=1. The overpowered aerosols support a revision of the global modern (2005) aerosol forcing 90 % CI to a narrower range [−1.0, −0.2] W m−2. The key parameter ECS in comparison to IPCC AR5 (and to the CMIP6 MME), the 90 % CI range is reduced from [1.5, 4.5] K ([2.0, 5.5] K) to [1.6, 2.4] K ([1.5, 2.2] K), with median value lowered from 3.0 K (3.7 K) to 2.0 K (1.8 K). Similarly we found for the transient climate response (TCR), the 90 % CI range shrinks from [1.0, 2.5] K ([1.2, 2.8] K) to [1.2, 1.8] K ([1.1, 1.6] K) and the median estimate decreases from 1.8 K (2.0 K) to 1.5 K (1.4 K). As often seen in other observational-based studies, the FEBE values for climate sensitivities are therefore somewhat lower but still consistent with those in IPCC AR5 and the CMIP6 MME. Using these parameters, we made projections to 2100 using both the Representative Concentration Pathway (RCP) and Shared Socioeconomic Pathway (SSP) scenarios, and compared them to the corresponding CMIP5 and CMIP6 multi-model ensembles (MMEs). The FEBE historical reconstructions (1880–2020) closely follow observations, notably during the 1998–2014 slowdown (“hiatus”). We also reproduce the internal variability with the FEBE and statistically validate this against centennial-scale temperature observations. Overall, the FEBE projections were 10 %–15 % lower but due to their smaller uncertainties, their 90 % CIs lie completely within the GCM 90 % CIs. This agreement means that the FEBE validates the MME, and vice versa.

Abstract One half of 2021s Nobel Physics prize was awarded to statistical physicist Giorgio Parisi and the other—the first in geophysics in 75 years—to climate scientists Syukoro Manabe and Klaus Hasselmann, the former for pioneering General Circulation Models and the latter (primarily) for proposing a statistical model explaining the climate as a slowly varying state driven by random weather noise. However, the Nobel committee recognized the climate laureates' work almost exclusively from the 1960s and 1970s. We update their report with the contributions from nonlinear geophysics and discuss the implications for the unity of geoscience, and for future climate modeling.

In Part 1, I considered the zero-dimensional heat equation, showing quite generally that conductive–radiative surface boundary conditions lead to half-ordered derivative relationships between surface heat fluxes and temperatures: the half-ordered energy balance equation (HEBE). The real Earth, even when averaged in time over the weather scales (up to ≈ 10 d), is highly heterogeneous. In this Part 2, the treatment is extended to the horizontal direction. I first consider a homogeneous Earth but with spatially varying forcing on both a plane and on the sphere: the new equations are compared with the canonical 1D Budyko–Sellers equations. Using Laplace and Fourier techniques, I derive the generalized HEBE (the GHEBE) based on half-ordered space–time operators. I analytically solve the homogeneous GHEBE and show how these operators can be given precise interpretations. I then consider the full inhomogeneous problem with horizontally varying diffusivities, thermal capacities, climate sensitivities, and forcings. For this I use Babenko's operator method, which generalizes Laplace and Fourier methods. By expanding the inhomogeneous space–time operator at both high and low frequencies, I derive 2D energy balance equations that can be used for macroweather forecasting, climate projections, and studying the approach to new (equilibrium) climate states when the forcings are all increased and held constant.

The original Budyko–Sellers type of 1D energy balance models (EBMs) consider the Earth system averaged over long times and apply the continuum mechanics heat equation. When these and the more phenomenological box models are extended to include time-varying anomalies, they have a key weakness: neither model explicitly nor realistically treats the conductive–radiative surface boundary condition that is necessary for a correct treatment of energy storage. In this first of a two-part series, I apply standard Laplace and Fourier techniques to the continuum mechanics heat equation, solving it with the correct radiative–conductive boundary conditions and obtaining an equation directly for the surface temperature anomalies in terms of the anomalous forcing. Although classical, this equation is half-ordered and not integer-ordered: the half-order energy balance equation (HEBE). A quite general consequence is that although Newton's law of cooling holds, the heat flux across surfaces is proportional to a half-ordered (not first-ordered) time derivative of the surface temperature. This implies that the surface heat flux has a long memory, that it depends on the entire previous history of the forcing, and that the temperature–heat flux relationship is no longer instantaneous. I then consider the case in which the Earth is periodically forced. The classical case is diurnal heat forcing; I extend this to annual conductive–radiative forcing and show that the surface thermal impedance is a complex valued quantity equal to the (complex) climate sensitivity. Using a simple semi-empirical model of the forcing, I show how the HEBE can account for the phase lag between the summer maximum forcing and maximum surface temperature Earth response. In Part 2, I extend all these results to spatially inhomogeneous forcing and to the full horizontally inhomogeneous problem with spatially varying specific heats, diffusivities, advection velocities, and climate sensitivities. I consider the consequences for macroweather (monthly, seasonal, interannual) forecasting and climate projections.

Atmospheric variability as a function of scale has been divided in various dynamical regimes with alternating increasing and decreasing fluctuations: weather, macroweather, climate, macroclimate, and megaclimate. Although a vast amount of data are available at small scales, the larger picture is not well constrained due to the scarcity and low resolution of long paleoclimatic time series. Using statistical techniques originally developed for the study of turbulence, we analyse the fluctuations of a centimetric-resolution dust flux time series from the EPICA Dome C ice core in Antarctica that spans the past 800 000 years. The temporal resolution ranges from annual at the top of the core to 25 years at the bottom, enabling the detailed statistical analysis and comparison of eight glaciation cycles and the subdivision of each cycle into eight consecutive phases. The unique span and resolution of the dataset allows us to analyse the macroweather and climate scales in detail. We find that the interglacial and glacial maximum phases of each cycle showed particularly large macroweather to climate transition scale τc (around 2 kyr), whereas mid-glacial phases feature centennial transition scales (average of 300 years). This suggests that interglacials and glacial maxima are exceptionally stable when compared with the rest of a glacial cycle. The Holocene (with τc≈7.9 kyr) had a particularly large τc, but it was not an outlier when compared with the phases 1 and 2 of other cycles. We hypothesize that dust variability at larger (climate) scales appears to be predominantly driven by slow changes in glaciers and vegetation cover, whereas at small (macroweather) scales atmospheric processes and changes in the hydrological cycles are the main drivers. For each phase, we quantified the drift, intermittency, amplitude, and extremeness of the variability. Phases close to the interglacials (1, 2, 8) show low drift, moderate intermittency, and strong extremes, while the “glacial” middle phases 3–7 display strong drift, weak intermittency, and weaker extremes. In other words, our results suggest that glacial maxima, interglacials, and glacial inceptions were characterized by relatively stable atmospheric conditions but punctuated by frequent and severe droughts, whereas the mid-glacial climate was inherently more unstable.

Abstract It was recently found that the accepted picture of atmospheric variability was in error by a large factor. Rather than being dominated by a series of narrow scale-range quasi-oscillatory processes with an unimportant white noise “background”, it turned out that the variance was instead dominated by a few wide range scaling processes albeit occasionally interspersed with superposed quasi-oscillations. Although the classical model implied that successive million year global temperature averages would differ by mere micro Kelvins, the implausibility had not been noticed. In contrast, the new picture inverts the roles of background and foreground and involves four (possibly five) wide range scaling processes. As with any new paradigm, there are consequences; in this paper we focus on the implications for the spectra, intermittency and the extremes. Intermittency is an expression of the spatio-temporal sparseness of strong events whereas the extremes refer to the tails of their probability distributions and both affect the spectra. Although we give some results for the macro and mega climate regimes, we focus on weather, macroweather and climate: from dissipation to Milankovitch scales.

At scales much longer than the deterministic predictability limits (about 10 days), the statistics of the atmosphere undergoes a drastic transition, the high-frequency weather acts as a random forcing on the lower-frequency macroweather. In addition, up to decadal and centennial scales the equivalent radiative forcings of solar, volcanic and anthropogenic perturbations are small compared to the mean incoming solar flux. This justifies the common practice of reducing forcings to radiative equivalents (which are assumed to combine linearly), as well as the development of linear stochastic models, including for forecasting at monthly to decadal scales. In order to clarify the validity of the linearity assumption and determine its scale range, we use last millennium simulations, with both the simplified Zebiak–Cane (ZC) model and the NASA GISS E2-R fully coupled GCM. We systematically compare the statistical properties of solar-only, volcanic-only and combined solar and volcanic forcings over the range of timescales from 1 to 1000 years. We also compare the statistics to multiproxy temperature reconstructions. The main findings are (a) that the variability in the ZC and GCM models is too weak at centennial and longer scales; (b) for longer than ≈ 50 years, the solar and volcanic forcings combine subadditively (nonlinearly) compounding the weakness of the response; and (c) the models display another nonlinear effect at shorter timescales: their sensitivities are much higher for weak forcing than for strong forcing (their intermittencies are different) and we quantify this with statistical scaling exponents.

In the present study, we investigate the scaling properties of the topography of Mars. Planetary topographic fields are well known to roughly exhibit (mono)fractal behavior. Indeed, the fractal formalism reproduces much of the variability observed in topography. Still, a single fractal dimension is not enough to explain the huge variability and intermittency. Previous studies have claimed that fractal dimensions might be different from one region to another, excluding a general description at the planetary scale. In this article, we analyze the Martian topographic data with a multifractal formalism to study the scaling intermittency. In the multifractal paradigm, the apparent local variation of the fractal dimension is interpreted as a statistical property of multifractal fields. We analyze the topography measured with the Mars Orbiter Laser altimeter (MOLA) at 300 m horizontal resolution, 1 m vertical resolution. We adapted the Haar fluctuation method to the irregularly sampled signal. The results suggest a multifractal behavior from the planetary scale down to 10 km. From 10 to 300 m, the topography seems to be simple monofractal. This transition indicates a significant change in the geological processes governing the Red Planet's surface.

On scales of ≈ 10 days (the lifetime of planetary-scale structures), there is a drastic transition from high-frequency weather to low-frequency macroweather. This scale is close to the predictability limits of deterministic atmospheric models; thus, in GCM (general circulation model) macroweather forecasts, the weather is a high-frequency noise. However, neither the GCM noise nor the GCM climate is fully realistic. In this paper we show how simple stochastic models can be developed that use empirical data to force the statistics and climate to be realistic so that even a two-parameter model can perform as well as GCMs for annual global temperature forecasts. The key is to exploit the scaling of the dynamics and the large stochastic memories that we quantify. Since macroweather temporal (but not spatial) intermittency is low, we propose using the simplest model based on fractional Gaussian noise (fGn): the ScaLIng Macroweather Model (SLIMM). SLIMM is based on a stochastic ordinary differential equation, differing from usual linear stochastic models (such as the linear inverse modelling – LIM) in that it is of fractional rather than integer order. Whereas LIM implicitly assumes that there is no low-frequency memory, SLIMM has a huge memory that can be exploited. Although the basic mathematical forecast problem for fGn has been solved, we approach the problem in an original manner, notably using the method of innovations to obtain simpler results on forecast skill and on the size of the effective system memory. A key to successful stochastic forecasts of natural macroweather variability is to first remove the low-frequency anthropogenic component. A previous attempt to use fGn for forecasts had disappointing results because this was not done. We validate our theory using hindcasts of global and Northern Hemisphere temperatures at monthly and annual resolutions. Several nondimensional measures of forecast skill – with no adjustable parameters – show excellent agreement with hindcasts, and these show some skill even on decadal scales. We also compare our forecast errors with those of several GCM experiments (with and without initialization) and with other stochastic forecasts, showing that even this simplest two parameter SLIMM is somewhat superior. In future, using a space–time (regionalized) generalization of SLIMM, we expect to be able to exploit the system memory more extensively and obtain even more realistic forecasts.

It was recently found that spectral solar incident flux (SIF) as a function of the ultraviolet wavelengths exhibits 1/f-type power-law correlations. In this study, an attempt was made to explore the residues of the SIF with respect to the Planck law over a wider range of wavelengths, from 115.5 to 629.5 nm. Using spectral, Haar and Detrended Fluctuation analyses, we show that over the range from 10–20 nm to the maximum lag (≈ 500 nm), the SIF residues have a scaling regime with fluctuation exponent H ≈ 0.37 but with high intermittency (C1 ≈ 0.16, multifractal index α≈ 1.7) and spectral exponent ≈ 1.46. Over the shorter wavelengths range we found on the contrary low intermittency (C1 ≈ 0) with spectral exponent ≈ 1 and H ≈ 0.

It is paradoxical that, while atmospheric dynamics are highly nonlinear and turbulent, atmospheric waves are commonly modelled by linear or weakly nonlinear theories. We postulate that the laws governing atmospheric waves are in fact high-Reynolds-number (Re), emergent laws so that – in common with the emergent high-Re turbulent laws – they are also constrained by scaling symmetries. We propose an effective turbulence–wave propagator which corresponds to a fractional and anisotropic extension of the classical wave equation propagator, with dispersion relations similar to those of inertial gravity waves (and Kelvin waves) yet with an anomalous (fractional) order Hwav/2. Using geostationary IR radiances, we estimate the parameters, finding that Hwav ≈ 0.17 ± 0.04 (the classical value = 2).

We present a selection of methodologies for using the palaeo-climate model component of the Coupled Model Intercomparison Project (Phase 5) (CMIP5) to attempt to constrain future climate projections using the same models. The constraints arise from measures of skill in hindcasting palaeo-climate changes from the present over three periods: the Last Glacial Maximum (LGM) (21 000 yr before present, ka), the mid-Holocene (MH) (6 ka) and the Last Millennium (LM) (850–1850 CE). The skill measures may be used to validate robust patterns of climate change across scenarios or to distinguish between models that have differing outcomes in future scenarios. We find that the multi-model ensemble of palaeo-simulations is adequate for addressing at least some of these issues. For example, selected benchmarks for the LGM and MH are correlated to the rank of future projections of precipitation/temperature or sea ice extent to indicate that models that produce the best agreement with palaeo-climate information give demonstrably different future results than the rest of the models. We also explore cases where comparisons are strongly dependent on uncertain forcing time series or show important non-stationarity, making direct inferences for the future problematic. Overall, we demonstrate that there is a strong potential for the palaeo-climate simulations to help inform the future projections and urge all the modelling groups to complete this subset of the CMIP5 runs.

We are used to the weather–climate dichotomy, yet the great majority of the spectral variance of atmospheric fields is in the continuous "background" and this defines instead a trichotomy with a "macroweather" regime in the intermediate range from ≈10 days to 10–30 yr (≈100 yr in the preindustrial period). In the weather, macroweather and climate regimes, exponents characterize the type of variability over the entire regime and it is natural to identify them with qualitatively different synergies of nonlinear dynamical mechanisms that repeat scale after scale. Since climate models are essentially meteorological models (although with extra couplings) it is thus important to determine whether they currently model all three regimes. Using last millennium simulations from four GCMs (global circulation models), we show that control runs only reproduce macroweather. When various (reconstructed) climate forcings are included, in the recent (industrial) period they show global fluctuations strongly increasing at scales > ≈10–30 yr, which is quite close to the observations. However, in the preindustrial period we find that the multicentennial variabilities are too weak and by analysing the scale dependence of solar and volcanic forcings, we argue that these forcings are unlikely to be sufficiently strong to account for the multicentennial and longer-scale temperature variability. A likely explanation is that the models lack important slow "climate" processes such as land ice or various biogeochemical processes.

High-resolution rainfall fields contain numerous zeros (i.e. pixels or time steps with no rain) which are either real or artificial – that is to say associated with the limit of detection of the rainfall measurement device. In this paper we revisit the enduring discussion on the source of this intermittency, e.g. whether it requires specific modelling. We first review the framework of universal multifractals (UM), which are commonly used to analyse and simulate geophysical fields exhibiting extreme variability over a wide range of scales with the help of a reduced number of parameters. However, this framework does not enable properly taking into account these numerous zeros. For example, it has been shown that performing a standard UM analysis directly on the field can lead to low observed quality of scaling and severe bias in the estimates of UM parameters. In this paper we propose a new simple model to deal with this issue. It is a UM discrete cascade process, where at each step if the simulated intensity is below a given level (defined in a scale invariant manner), it only has a predetermined probability to survive and is otherwise set to zero. A threshold can then be implemented at the maximum resolution to mimic the limit of detection of the rainfall measurement device. While also imperfect, this simple model enables explanation of most of the observed behaviour, e.g. the presence of scaling breaks, or the difference between statistics computed for single events or longer periods.

Geophysical processes are typically variable over huge ranges of space-time scales. This has lead to the development of many techniques for decomposing series and fields into fluctuations Δv at well-defined scales. Classically, one defines fluctuations as differences: (Δvdiff = v(x+Δx)-v(x) and this is adequate for many applications (Δx is the "lag"). However, if over a range one has scaling Δv &propto; ΔxH, these difference fluctuations are only adequate when 0 < H < 1. Hence, there is the need for other types of fluctuations. In particular, atmospheric processes in the "macroweather" range &approx;10 days to 10–30 yr generally have −1 < H < 0, so that a definition valid over the range −1 < H < 1 would be very useful for atmospheric applications. A general framework for defining fluctuations is wavelets. However, the generality of wavelets often leads to fairly arbitrary choices of "mother wavelet" and the resulting wavelet coefficients may be difficult to interpret. In this paper we argue that a good choice is provided by the (historically) first wavelet, the Haar wavelet (Haar, 1910), which is easy to interpret and – if needed – to generalize, yet has rarely been used in geophysics. It is also easy to implement numerically: the Haar fluctuation (ΔvHaar at lag Δx is simply equal to the difference of the mean from x to x+ Δx/2 and from x+Δx/2 to x+Δx. Indeed, we shall see that the interest of the Haar wavelet is this relation to the integrated process rather than its wavelet nature per se. Using numerical multifractal simulations, we show that it is quite accurate, and we compare and contrast it with another similar technique, detrended fluctuation analysis. We find that, for estimating scaling exponents, the two methods are very similar, yet Haar-based methods have the advantage of being numerically faster, theoretically simpler and physically easier to interpret.

Lindborg et al. (2010) claim that the apparent spectrum power law E(k) ≈ k−3 on scales ≥600 km obtained with the help of commercial jetliner trajectory deviations (GASP and Mozaic databases) could not be brought into question (Lovejoy et al., 2009a), because this spectrum corresponds to "a well known theory of quasi-geostrophic turbulence developed by Charney (1971)". Lindborg et al. (2010) also claim that "limitations [of this theory] have been relaxed in many of the modern models of atmospheric turbulence". We show that both claims are irrelevant and that generalized scale invariance (GSI) is indispensable to go beyond the quasi-geostrophic limitations, to go in fact from scale analysis to scaling analysis in order to derive better analytical models. In this direction, we derive vorticity equations in a space of (fractal) dimension D=2+Hz (0 ≤ Hz ≤ 1), which corresponds to a first step in the derivation of a dynamical alternative to the quasi-geostrophic approximation and turbulence. The corresponding precise definition of fractional dimensional turbulence already demonstrates that the classical 2-D and 3-D turbulence are not the main options to understand atmospheric dynamics. Although (2 + Hz)-D turbulence (with 0 < Hz < 1) has more common features with 3-D turbulence than with 2-D turbulence, it has nevertheless very distinctive features: its scaling anisotropy is in agreement with the layered pancake structure, which is typical of rotating and stratified turbulence but not of the classical 3-D turbulence.

By direct statistical analysis we show that over almost all their range of scales and to within typically better than ±1%, atmospheric fields obtained from analyses and numerical integrations of atmospheric models have the multifractal structure predicted by multiplicative cascade models. We quantify this for the horizontal wind, temperature, and humidity fields at 5 different pressure levels for the ERA40 reanalysis, the Canadian Meteorological Centre Global Environmental Multiscale (CMC, GEM) model, as well as the National Oceanographic and Atmospheric Administration Global Forecasting System (NOAA, GFS). We investigate the additional prediction that the cascade belongs to a universal multifractal basin of attraction. By demonstrating a "Levy collapse" of the statistical moments to within ±2 to ±5% over most of the range of scales, we conclude that there is good evidence for this. Finally, we discuss how this stochastic multiplicative cascade structure can be exploited in improving ensemble forecasts.

Due to both systematic and turbulent induced vertical fluctuations, the interpretation of atmospheric aircraft measurements requires a theory of turbulence. Until now virtually all the relevant theories have been isotropic or "quasi isotropic" in the sense that their exponents are the same in all directions. However almost all the available data on the vertical structure shows that it is scaling but with exponents different from the horizontal: the turbulence is scaling but anisotropic. In this paper, we show how such turbulence can lead to spurious breaks in the scaling and to the spurious appearance of the vertical scaling exponent at large horizontal lags. We demonstrate this using 16 legs of Gulfstream 4 aircraft near the top of the troposphere following isobars each between 500 and 3200 km in length. First we show that over wide ranges of scale, the horizontal spectra of the aircraft altitude are nearly k-5/3. In addition, we show that the altitude and pressure fluctuations along these fractal trajectories have a high degree of coherence with the measured wind (especially with its longitudinal component). There is also a strong phase relation between the altitude, pressure and wind fluctuations; for scales less than ≈40 km (on average) the wind fluctuations lead the pressure and altitude, whereas for larger scales, the pressure fluctuations leads the wind. At the same transition scale, there is a break in the wind spectrum which we argue is caused by the aircraft starting to accurately follow isobars at the larger scales. In comparison, the temperature and humidity have low coherencies and phases and there are no apparent scale breaks, reinforcing the hypothesis that it is the aircraft trajectory that is causally linked to the scale breaks in the wind measurements. Using spectra and structure functions for the wind, we then estimate their exponents (β, H) at small (5/3, 1/3) and large scales (2.4, 0.73). The latter being very close to those estimated by drop sondes (2.4, 0.75) in the vertical direction. In addition, for each leg we estimate the energy flux, the sphero-scale and the critical transition scale. The latter varies quite widely from scales of kilometers to greater than several hundred kilometers. The overall conclusion is that up to the critical scale, the aircraft follows a fractal trajectory which may increase the intermittency of the measurements, but doesn't strongly affect the scaling exponents whereas for scales larger than the critical scale, the aircraft follows isobars whose exponents are different from those along isoheights (and equal to the vertical exponent perpendicular to the isoheights). We bolster this interpretation by considering the absolute slopes (|Δz/Δx|) of the aircraft as a function of lag Δx and of scale invariant lag Δx/Δz1/Hz. We then revisit four earlier aircraft campaigns including GASP and MOZAIC showing that they all have nearly identical transitions and can thus be easily explained by the proposed combination of altitude/wind in an anisotropic but scaling turbulence. Finally, we argue that this reinterpretation in terms of wide range anisotropic scaling is compatible with atmospheric phenomenology including convection.

The classical models of volcanic eruptions assume that they originate as a consequence of critical stresses or critical strain rates being exceeded in the magma followed by catastrophic fragmentation. In a recent paper (Gaonac'h et al., 2003) we proposed an additional mechanism based on the properties of complex networks of overlapping bubbles; that extreme multibubble coalescence could lead to catastrophic changes in the magma rheology at a critical vesicularity. This is possible because at a critical vesicularity Pc (the percolation threshold), even in the absence of external stresses the magma fragments. By considering 2-D percolation with the (observed) extreme power law bubble distributions, we showed numerically that P2c had the apparently realistic value ≈0.7. The properties of percolating systems are, however, significantly different in 2-D and 3-D. In this paper, we discuss various new features relevant to 3-D percolation and compare the model predictions with empirical data on explosive volcanism. The most important points are a) bubbles and magma have different 3-D critical percolation points; we show numerically that with power law bubble distributions that the important magma percolation threshold P3c,m has the high value ≈0.97±0.01, b) a generic result of 3-D percolation is that the resulting primary fragments will have power law distributions with exponent B3f≈1.186±0.002, near the empirical value (for pumice) ≈1.1±0.1; c) we review the relevant percolation literature and point out that the elastic properties may have lower – possibly more realistic – critical vesicularities relevant to magmas; d) we explore the implications of long range correlations (power law bubble distributions) and discuss this in combination with bubble anisotropy; e) we propose a new kind of intermediate "elliptical" dimensional percolation involving differentially elongated bubbles and show that it can lead to somewhat lower critical thresholds. These percolation mechanisms for catastrophically weakening magma would presumably operate in conjunction with the classical critical stress and critical strain mechanisms. We conclude that percolation theory provides an attractive theoretical framework for understanding highly vesicular magma.

Starting about thirty years ago, new ideas in nonlinear dynamics, particularly fractals and scaling, provoked an explosive growth of research both in modeling and in experimentally characterizing geosystems over wide ranges of scale. In this review we focus on scaling advances in solid earth geophysics including the topography. To reduce the review to manageable proportions, we restrict our attention to scaling fields, i.e. to the discussion of intensive quantities such as ore concentrations, rock densities, susceptibilities, and magnetic and gravitational fields. We discuss the growing body of evidence showing that geofields are scaling (have power law dependencies on spatial scale, resolution), over wide ranges of both horizontal and vertical scale. Focusing on the cases where both horizontal and vertical statistics have both been estimated from proximate data, we argue that the exponents are systematically different, reflecting lithospheric stratification which – while very strong at small scales – becomes less and less pronounced at larger and larger scales, but in a scaling manner. We then discuss the necessity for treating the fields as multifractals rather than monofractals, the latter being too restrictive a framework. We discuss the consequences of multifractality for geostatistics, we then discuss cascade processes in which the same dynamical mechanism repeats scale after scale over a range. Using the binomial model first proposed by de Wijs (1951) as an example, we discuss the issues of microcanonical versus canonical conservation, algebraic ("Pareto") versus long tailed (e.g. lognormal) distributions, multifractal universality, conservative and nonconservative multifractal processes, codimension versus dimension formalisms. We compare and contrast different scaling models (fractional Brownian motion, fractional Levy motion, continuous (in scale) cascades), showing that they are all based on fractional integrations of noises built up from singularity basis functions. We show how anisotropic (including stratified) models can be produced simply by replacing the usual distance function by an anisotropic scale function, hence by replacing isotropic singularities by anisotropic ones.

We investigate the statistical properties of dendritic drainage areas from diverse geological environments (Deception Canyon, Utah and the Loess Plateau, China) using narrow band visible ASTER satellite images. We show that from 240 m to 7680 m, the isotropic (angle integrated) energy spectra E(k) of all the fields closely follow a power law form: E(k)∝k−β where k is a wave number and β a scale invariant exponent. In spite of this good isotropic scaling, images with very similar β's and similar isotropic multifractal exponents have distinct textures; we suggest that the differences are primarily due to anisotropy, which is nevertheless scaling. We develop the new "Differential Anisotropy Scaling" technique to characterize this scale-by-scale (differential) anisotropy and we test it on simulated anisotropic scaling fields. The method gives useful characterizations of the scale by scale anisotropy irrespective of whether or not the analyzed field is scaling. When the anisotropy is not too strong, the parameters can be interpreted as scale invariant anisotropy exponents. Viewed as a method of estimating these exponents, it has the advantage of relying on two linear regressions rather than on complex higher dimensional nonlinear ones. When applied to dendritic drainage basins we find that they have distinct anisotropies characterized by differential anisotropy stretching and rotation parameters as well as by a distinct absolute anisotropy at the reference scale of 960 m. Our new method allows us to statistically distinguish, not only between two geologically different drainage basins (the China Loess Plateau and Utah Deception Canyon), but also between different regions of the same China drainage system.

This paper shows how modern ideas of scaling can be used to model topography with various morphologies and also to accurately characterize topography over wide ranges of scales. Our argument is divided in two parts. We first survey the main topographic models and show that they are based on convolutions of basic structures (singularities) with noises. Focusing on models with large numbers of degrees of freedom (fractional Brownian motion (fBm), fractional Levy motion (fLm), multifractal fractionally integrated flux (FIF) model), we show that they are distinguished by the type of underlying noise. In addition, realistic models require anisotropic singularities; we show how to generalize the basic isotropic (self-similar) models to anisotropic ones. Using numerical simulations, we display the subtle interplay between statistics, singularity structure and resulting topographic morphology. We show how the existence of anisotropic singularities with highly variable statistics can lead to unwarranted conclusions about scale breaking. We then analyze topographic transects from four Digital Elevation Models (DEMs) which collectively span scales from planetary down to 50 cm (4 orders of magnitude larger than in previous studies) and contain more than 2×108 pixels (a hundred times more data than in previous studies). We use power spectra and multiscaling analysis tools to study the global properties of topography. We show that the isotropic scaling for moments of order ≤2 holds to within ±45% down to scales ≈40 m. We also show that the multifractal FIF is easily compatible with the data, while the monofractal fBm and fLm are not. We estimate the universal parameters (α, C1) characterizing the underlying FIF noise to be (1.79, 0.12), where α is the degree of multifractality (0≤α≤2, 0 means monofractal) and C1 is the degree of sparseness of the surface (0≤C1, 0 means space filling). In the same way, we investigate the variation of multifractal parameters between continents, oceans and continental margins. Our analyses show that no significant variation is found for (α, C1) and that the third parameter H, which is a degree of smoothing (higher H means smoother), is variable: our estimates are H=0.46, 0.66, 0.77 for bathymetry, continents and continental margins. An application we developped here is to use (α, C1) values to correct standard spectra of DEMs for multifractal resolution effects.

In this paper, we present evidence that intermittency of Eulerian and Lagrangian turbulence of ocean temperature and plankton fields is multifractal and furthermore can be analysed with the help of universal multifractals. We analyse time series of temperature and in vivo fluorescence taken from a drifter in the mixed coastal waters of the eastern English Channel. Two analysis techniques are used to compute the fundamental universal multifiractal parameters, which describe all the statistics of the turbulent fluctuations: the analysis of the scale invariant structure function exponent ζ(q) and the Double Trace Moment technique. At small scales, we do not detect any significant difference between the universal multifiractal behavior of temperature and fluorescence in an Eulerian framework. This supports the hypothesis that the latter is passively advected with the flow as the former. On the one hand, we show that large scale measurements are Lagrangian and indeed we obtain for temperature fluctuations a ω2 power spectrum corresponding to the theoretical scaling of a Lagrangian passive scalar. Furthermore, we show that Lagrangian temperature fluctuations are multiscaling and intermittent. On the other hand, the flatter slope at large scales of the fluorescence power spectrum points out that the plankton is at these scales a 'biologically active' scalar.

Over wide ranges of scale, orographic processes have no obvious scale; this has provided the justification for both deterministic and monofractal scaling models of the earth's topography. These models predict that differences in altitude (Δh) vary with horizontal separation (l) as Δh ≈ lH. The scaling exponent has been estimated theoretically and empirically to have the value H=1/2. Scale invariant nonlinear processes are now known to generally give rise to multifractals and we have recently empirically shown that topography is indeed a special kind of theoretically predicted "universal" multifractal. In this paper we provide a multifractal generalization of the l1/2 law, and propose two distinct multifractal models, each leading via dimensional arguments to the exponent 1/2. The first, for ocean bathymetry assumes that the orographic dynamics are dominated by heat fluxes from the earth's mantle, whereas the second - for continental topography - is based on tectonic movement and gravity. We test these ideas empirically on digital elevation models of Deadman's Butte, Wyoming.

We empirically investigate the scaling behaviour of the horizontal wind along the vertical direction using 287 radiosonde soundings with a resolution of 50 m. We compare the results obtained with those of the horizontal temporal behaviour in the framework of Generalized Scaling Invariance and the unified Scaling model of atmospheric dynamics. We find the scaling to be very well respected over the range 50 m - 13 km (nearly the entire troposphere) and we estimate the universal multifractal indices which characterize the statistics in the vertical. By comparing our result with those obtained in the horizontal we show that the degree of stratification is different for mean and extreme structures. Finally, we theoretically discuss the necessary improvements to the Unified Multifractal model needed to account for them.