U52A-01 INVITED
Tokunaga fractals: Stream networks, DLA, aftershocks, and biology
A Tokunaga fractal is a self-similar branching topology that includes both primary and side branching. It was developed in terms of the self-similar branching of stream networks. It was subsequently shown that the branching topology of diffusion limited aggregation (DLA) clusters is also a Tokunaga fractal. The epidemic type aftershock sequence (ETAS) model has been widely used to model the statistics of seismicity. A self-similar version of ETAS is the branching aftershock sequence (BASS) model (GRL, 34, L12303,2007). The aftershocks of the main shock represent a primary branching structure. The aftershocks of aftershocks represent the side- branching structure. We show that deterministic BASS simulations give a Tokunaga fractal structure. The primary branching ratio is given aby the b-value of the aftershocks. The characteristic earthquake magnitude is given by the modified form of Bath's law. The average branching ratio of all parent-daughter is unity. Tokunaga fractals can also be found in biology. A specific example is the vein structure of leaves.
U52A-02
Fractal scaling for surface water-subsurface water interaction through the Earths crust
Landscape topography from mountain ranges to the smallest hills induces the pressure boundary condition at ground surface that controls groundwater circulation. This interplay between surface water and groundwater controls the circulation patterns of deep groundwater in the Earth's crust, the water balance in watersheds, as well as solute transport from the continents to the oceans. Separating the topography in a Fourier spectrum both represents the fractal ground surface topography in fluvial and glacial landscapes and provides an exact solution for the three-dimensional groundwater flows including the surface water interaction. In boreal landscapes where the groundwater surface follows the topography, all landscape features have a significant impact on the surface–subsurface water interaction. However, because of the decaying permeability with depth there is a clear tendency that the interfacial flux tends to be dominated by small-scale features, while the flux through deeper subsurface flow paths tends to be controlled by larger-scale features. In the order of 10 % of the precipitation in Scandinavia infiltrates the ground, but only about 1 ‰ of the groundwater recharge reaches deeper than 400 m. The fractal nature of surface-subsurface water fluxes yields a scale-independent distribution of subsurface water residence times for both near-surface fluvial systems and deeper hydrogeological flows. This holds for small- scale exchange occurring in the stream-beds and the deeper circulation of the Earths crust. The fact that we can now estimate the renewal rate of deep groundwater based on a spectral representation of topography over the entire continent has far-reaching implications for the long-term management of groundwater resources. The improved understanding of the exchange processes in the hyporheic zone and its interaction with biogeochemistry and colloids is of utmost importance for understanding recent environmental problems like eutrophication from nitrogen and phosphorus and spreading of toxic substances like heavy metals. http://www.lwr.kth.se
U52A-03 INVITED
SCALING IN ECOSYSTEMS AND THE LINKAGE OF MACROECOLOGICAL LAWS
Are there predictable linkages among macroecological laws regulating size and abundance of organisms that are ubiquitously supported by empirical observations and that ecologists treat traditionally as independent? Do fragmentation of habitats, or reduced supply of energy and matter, result in predictable changes on whole ecosystems as a function of their size? Using a coherent theoretical framework based on scaling theory, it is argued that the answer to both these questions is affirmative. The concern of the talk is with the comparatively simple situation of the steady state behavior of a fully developed ecosystem in which, over evolutionary time, resources are exploited in full, individual and collective metabolic needs are met and enough time has elapsed to produce a rough balance between speciation and extinction and ecological fluxes. While ecological patterns and processes often show great variation when viewed at different scales of space, time, organismic size and organizational complexity, there is also widespread evidence for the existence of scaling regularities as embedded in macroecological "laws" or rules. These laws have commanded considerable attention from the ecological community. Indeed they are central to ecological theory as they describe the features of complex adaptive systems shown by a number of biological systems, and perhaps for the investigation of the dynamic origin of scale invariance of natural forms in general. The species-area and relative species-abundance relations, the scaling of community and species' size spectra, the scaling of population densities with their mean body mass and the scaling of the largest organism with ecosystem size are examples of such laws. Borrowing heavily from earlier successes in physics, it will be shown how simple mathematical scaling arguments, following from dimensional and finite-size scaling analyses, provide theoretical predictions of the inter- relationships among the species abundance relationship, the species-area relationship and community size spectra, in excellent accord with empirical data. The main conclusion is that the proposed scaling framework, along with the questions and predictions it provides, serves as a starting point for a novel approach to macroecological analysis.
U52A-04
Analysis and Forecasting of Shoreline Position
Analysis of historical shoreline positions on sandy coasts, in the geologic record, and study of sea-level rise curves reveals that the dynamics of the underlying processes produce temporal/spatial signals that exhibit power scaling and are therefore self-affine fractals. Self-affine time series signals can be quantified over many orders of magnitude in time and space in terms of persistence, a measure of the degree of correlation between adjacent values in the stochastic portion of a time series. Fractal statistics developed for self-affine time series are used to forecast a probability envelope bounding future shoreline positions. The envelope provides the standard deviation as a function of three variables: persistence, a constant equal to the value of the power spectral density when 1/period equals 1, and the number of time increments. The persistence of a twenty-year time series of the mean-high-water (MHW) shoreline positions was measured for four profiles surveyed at Duck, NC at the Field Research Facility (FRF) by the U.S. Army Corps of Engineers. The four MHW shoreline time series signals are self-affine with persistence ranging between 0.8 and 0.9, which indicates that the shoreline position time series is weakly persistent (where zero is uncorrelated), and has highly varying trends for all time intervals sampled. Forecasts of a probability envelope for future MHW positions are made for the 20 years of record and beyond to 50 years from the start of the data records. The forecasts describe the twenty-year data sets well and indicate that within a 96% confidence envelope, future decadal MHW shoreline excursions should be within 14.6 m of the position at the start of data collection. This is a stable-oscillatory shoreline. The forecasting method introduced here includes the stochastic portion of the time series while the traditional method of predicting shoreline change reduces the time series to a linear trend line fit to historic shoreline positions and extrapolated linearly to forecast future positions with a linearly increasing mean that breaks the confidence envelope eight years into the future and continues to increase. The traditional method is a poor representation of the observed shoreline position time series and is a poor basis for extrapolating future shoreline positions.
U52A-05
Spatial Variability of Multifractality of Exploratory Geochemical Maps
Exploratory geochemical maps created from concentration values of trace elements in the earth surface media (rocks, tills, soils, stream sediments and water) for mineral exploration and environment impact studies often show singularities that can be mathematically described by means of fractal and multifractal models. Implementation of multifractal modeling in various scales (entire map, sub-areas of maps and pixels) can characterize localized multifractality of geochemical maps which have been proved useful for description of underlying geo-processes responsible to the spatial variance of geochemical fields. This paper introduces several indexes of local singularity and multifractality that have been proposed for mapping spatial variability of geochemical fields and for delineating anomalous areas for prediction of undiscovered mineral deposits. Average singularity and variability of singularity over entire map, sub-area and pixels are calculated from exploratory data and compared for several mineral deposits types in various mineral districts. The results show that the estimated values of local singularity and its variability within a small area can be used to characterize geochemical fields related to mineralization. As a case study, maps at 2km spatial resolution created from concentration values of twelve trace elements Au, As, Sb, Sn, Zn, Pb, Cd, and Ag etc. in more than 25,000 stream sediment from Yuannan Province, China were used in this research. It shows that hydrothermal mineralization processes corresponding to gold, lead/zinc, copper, tin and nickel etc. in the Yunnan Province generally correspond to local singularity in geochemical data in stream sediments. Identification of mineralization caused local singularity using multifractal models can assist in mapping anomalies for prediction of mineral deposits.
U52A-06 INVITED
The Turbulent Structure of the Atmosphere: Vorticity, Winds and Temperature Emerge From Molecular Motion
Application of generalized scale invariance to horizontal airborne observations of winds, temperature, ozone and humidity reveals the atmosphere as a random, non-Gaussian Levy process, having mean scaling exponents H (conservation), C1 (intermittency) and alpha (Levy) of 0.56, 0.05 and 1.6 respectively in the cases of winds and temperature. A correlation between the intermittency of temperature and the ozone photodissociation rate in the Arctic lower stratosphere is interpreted in terms of the ring currents of non-equilibrium statistical mechanics in which vortices, fluid dynamical behavior, emerge from thermalized populations of Maxwellian molecules subjected to an anisotropy in the form of a flux. The emergence of jet streams and the definition of atmospheric temperature are examined in the light of these results. The vertical scaling of wind, temperature and humidity is examined through the depth of the troposphere using data observed by GPS dropsondes from the NOAA Gulfstream 4 aircraft over the eastern Pacific Ocean in boreal winter. The results exclude isotropic turbulence in the atmosphere, and reveal the structure of static, moist static and dynamic (Richardson number) stabilities to be sparse fractal sets. Each stable layer contains a set of smaller scale unstable sublayers, each of which in turn contains a set of stable sub-sublayers and so on. The moist static stability scales differently to the dry static stability in the lower troposphere. As with the ‘horizontal' data, the ‘vertical' data reveal a correlation between H for horizontal wind and measures of jet stream strength. It is pointed out that these results provide potentially a new way of testing numerical models of the atmosphere.
U52A-07
The remarkable wide range space-time scaling of atmospheric reflectivities and visible, infra red and microwave radiances
Numerous studies have shown that over various ranges, atmospheric fields are scaling. However until now, for various technical reasons they have been restricted to relatively small ranges of scale, and to relatively small data sets (with few realizations). In particular, the key question as to the large scale space-time limits of scaling has not been empirically established. Does the scaling extend up to planetary scales? What are the corresponding long time scales? Is space-time differentially stratified or roughly isotropic? In this presentation we answer these questions using the first truly global scale analysis of satellite radiances examining 14 passive channels ranging from visible to infra red to passive microwave as well as (active) radar reflectivities. The spatial resolutions depend somewhat on wavelength but are from several kilometers up to planetary scales (20000km), and from (typically) 2- 4 days in time up to 10 years. Two basic analyses were performed: the first purely spatial analysis at full spatial resolution (but orbit by orbit), the second in space-time using a lower resolution (100 km, 2 – 4 day resolution) space-time grid. The spatial analysis showed that without exception the radiances are remarkably scale invariant displaying statistics typically within ±1 - ±2% of those predicted for multifractal cascades. The high accuracy with which scaling holds makes it one of the most accurately obeyed laws in meteorology. For the wavelengths ≤1.6 cm (passive microwave, visible, near and far infra red), this was true over ranges which extend from 5000 - 10000 km down to the resolution scale (≈ 4 km) with effective outer cascade scales in the range 8000 - 20000 km. For these wavelengths, the statistics were near those of turbulent passive scalars. In comparison, for both the passive microwave at 3 cm and the active microwave channel (precipitation radar, 2.2 cm), the cascade was much sparser and the outer scale was around 32000 km indicating that even at planetary scales there was residual variability due to interaction with other processes. In all cases, the outer scale was about twice as large in the east-west direction compared to the north-south direction. Finally, the space-time analyses on the gridded data showed that for the passive scalar - like radiances, the scaling extends to about 22 days (the lifetime of the largest planetary scale structures). The resulting space-time transformation implies a velocity of about 6m/s, and also that space-time is not differentially stratified. We note that for the (precipitation) reflectivity data - the scaling extends to at least 190 days and implies a space-time transformation with a much larger smaller velocity (50 cm/s). Since the data cover the principle wavebands responsible for incoming and outgoing radiative fluxes, and since the latter are strongly nonlinearly coupled with the atmosphere, it is hard to avoid the conclusion that the atmosphere is a scale invariant dynamical process.
U52A-08
Temporal Scaling of Biogeochemical Reaction Rates
In at least two disparate areas of organic and inorganic geochemistry---the microbial degradation of detritus and the dissolution of minerals in sediments and soils---apparent rate constants k have been observed to diminish with the "age" t of the substrate like k(t) ~eq a t-b, where a ~eq 0.2 and b ~eq 1.0. Published reports display up to ten orders of magnitude in time [1,2]. Because the accuracy of biogeochemical models typically depends crucially on the specification of such rates, an understanding of this scaling law has important implications for predicting the evolution of biogeochemical cycles, especially the cycles of carbon and oxygen. The power-law decay of rates likely derives from a combination of chemical and physical heterogeneity. In a purely chemical scenario, an intrinsically heterogeneous substrate (e.g., a mixture of organic matter ranging from "labile" to "recalcitrant") is assumed to produce the observed slowdown of k(t). In contrast, a physical model assumes a homogeneous substrate in which rates nevertheless vary microscopically due to spatially varying physical constraints. Here we consider the extreme case of a purely physical origin and test its consistency with observations [3]. We first show how a diffusion-limited reaction-diffusion system leads to a logarithmic decay of the substrate. We then show how the power-law for k(t) derives from this logarithmic decay. We obtain not only the correct exponent b=1 but also a good approximation of the prefactor a. By constructing an extensive database of previously published measurements, we show that observations compare well to predictions. The particular way in which diffusion-limitation manifests itself varies from problem to problem. In the case of detrital decay in sediments, we suggest that rates are limited by contact of substrate with extracellular enzymes [3]. Mechanisms in soils are likely similar. For mineral dissolution is sediments, we suggest that rates are limited by diffusion of reactants from the seafloor. [1] J.~J.~Middelburg, Geochim.~Cosmochim.~Acta 53, 1577 (1989). [2] K.~Maher, D.~J.~DePaolo, J.~C.-F.~Lin, Geochim.~Cosmochim.~Acta 68, 4629. [3] D.~H.~Rothman and D.~C.~Forney, Science 316, 1325 (2004).
U52A-09
Seasonality Effects on Nonlinear Properties of Hydrometeorological Records: A New Method of Data Analysis
Climatic time series in general, and hydrological time series in particular, exhibit pronounced annual periodicity. This periodicity and its corresponding harmonics affect the nonlinear properties of the relevant time series (i.e., the long-range volatility correlations and width of multifractal spectrum) and thus have to be filtered out before studying fractal and volatility properties. We compare several filtering techniques (one of them proposed here) and find that in order to eliminate the periodicity effect on the nonlinear properties of the time series (i.e., the volatility and multifractal properties) it is necessary to filter out the seasonal standard deviation in addition to the filtering of the seasonal mean. The obtained results indicate weak volatility correlations (weak nonlinearity) in the river data, and this can be seen using different filterings approaches. [1] Livina~V.~N., Y.~Ashkenazy, A.~Bunde, and S.~Havlin, Seasonality effects on nonlinear properties of hydrometeorological records, in Extremes, Trends, and Correlations in Hydrology and Climate (ed. by J.P.Kropp & H.-J.Schellnhuber), Springer, Berlin, submitted.
U52A-10 INVITED
Multiscale Dynamics of the Magnetosphere
The dynamics of the magnetosphere is driven by the turbulent solar wind, and exhibits complex behavior with global, regional and local features. The global features are in general captured by the geomagnetic indices and the regional and local features are measured by spacecraft-based imagers and ground-based instruments. The global dynamical behavior has been studied extensively using nonlinear dynamical techniques. However the presence of a wide range of scales limits the ability of these techniques and a mean-field approach is used to obtain a deterministic model. This description is further improved by introducing a weighted averaging, with the weights determined by the distribution of sates in the reconstructed phase space. The multiscale aspects can not be described within a deterministic framework and a Bayesian approach is used to compute the conditional probabilities from the correlated solar wind - magnetosphere data. The regional features of the dynamics is studied using the mutual information functions computed from the magnetic field data from the high latitude magnetometer stations. The spreads in the average mutual information show a good correlation with the solar wind convective electric field and sudden changes in the dynamic pressure. The distribution of scales in the magnetosphere is studied using an extensive database. The distributions of the waiting times deviate significantly from a power law as well as stretched exponential distributions, and show a scaling with respect to the mean, indicating a limited role of long-term correlations in the magnetospheric dynamics.