U43B-1122
Is source distribution of radiogenic heat generation inside the Earth's crust fractal?
The crustal radiogenic heat sources contribute to about 40% of the heat flow at the surface in continental regions making this an important element in determining the crustal temperature distribution. Most of the radioactive isotopes are concentrated in the upper continental crust. In the absence of direct measurements of radioactive heat generation, various models for the depth distribution of radiogenic heat sources have been proposed in the literature out of which the exponential model is being widely used. However, we observed that variation of heat production rate in granites with depth in the three boreholes GPK1 & GPK2 (European Hot-Dry-Rock site in Soultz) and KTB, Germany is not exponential. Deviation from exponential model is evident at greater depths. We therefore argue that heat production in crust cannot be explained by a simple exponential model. Another opportunity to study more directly the variation of heat production with depth is provided by the lower crustal data from Western Canterbury region, New Zealand. At first look the heat production rate appeared to vary randomly between 0.92 and 2.39 micro w/m3 with the mean value of 1.56 micro w/m3, which is very much higher than the normal value taken for the lower crust assuming exponential model. This contributes to an important evidence for the deviation of the exponential model of heat production rate of lower crustal rocks. Study of rate of decrease in heat production with depth is necessary to consider the variations of heat flow with depth and heat flow extrapolation to greater depth is crucial to obtain the boundary conditions in various numerical models. Hence, to find an acceptable solution we carried out rescaled range analysis of the calculated heat production data of Soultz boreholes to understand the distribution. Value of Hurst coefficient obtained by rescaled range analysis of the data is around 0.84, which indicates that the radiogenic source distribution is more complex than a simple exponential model and may follow a fractal distribution. Hence, it may be necessary to employ the concept of fractal behaviour of these quantities in 1-D steady state heat conduction equation to compute the crustal and lithospheric temperature distributions.
U43B-1123
How Fractal are Coastlines Really? Observation and Theory
Rocky coastlines have been held up as a prime example of fractal geometry since Mandelbrot introduced the concept. However, we will present a map of the fractal dimensions measured for the contiguous United States coastline which shows that many open-ocean sand—and even rocky—coastlines have fractal dimensions close to one; i.e. they tend to not be very fractal. The fractal nature of rocky coastlines likely represents an inherited fluvial or glacial signature that tends to be erased by coastal processes. Recent theoretical and numerical-modeling developments indicate that wave-driven coastal processes on sandy shores tend to produce one-dimensional coastlines. Gradients in alongshore sediment flux tend to smooth a shoreline, as long as the local wave climate is dominated by ‘low-angle' waves (waves that approach the coastline in deep water from angles, relative to the coastline orientation, that are lower than the sediment-flux- maximizing angle). Even when a regional wave climate is dominated by high-angle waves—which produce an instability in plan-view shoreline shape—on the large scale, coastlines self organize in a way that produces locally low-angle-dominated wave climates almost everywhere. These processes explain why wave-dominated sandy coastlines, such as the Carolina and Texas coasts, exhibit fractal dimensions barely above one; wave- driven alongshore transport is an anti-fractal landsculpting agent over a range of scales greater than 0.2 km. In contrast, fluvial landsculpting produces famously fractal topography. When rapid sea-level rise causes the approximately horizontal plane of sea level to intersect a fractal fluvial topography, a fractal coastline results. Where wave energy is low, relative to rock erodibility, the fluvial fractal signature can persist. However, on the rocky West Coast of the US, fractal dimensions are relatively low (1.1 – 1.2), suggesting modification by wave-driven processes; that the production and rearrangement of sediment into ever-expanding pocket beaches has been reducing the fractality of this high-wave-energy, relatively easily eroded coastline. Glacially carved coastlines, such as that of Maine (and some parts of western Britain and Norway), exhibit high fractal dimensions (approximately 1.5), where erodibility is low enough the self-similarity of the intersection of sea-level with a glacially sculpted topography remains. Although wave-driven coastal processes tend to generate low-fractal-dimension shorelines, on sandy coastlines dominated by tidal currents, coastal processes also etch a fractal dendritic network of channels into the coastline. Tidally dominated coastlines, such as those in the Georgia Bight (Southeastern US), sport highly fractal shapes as a result (fractal dimensions approximately 1.5).
U43B-1124
A new theoretical treatment of dispersion (in absence of diffusion) using percolation theory.
We argue that in porous media with sufficiently wide pore-size distributions to require use of critical path analysis in upscaling K, the tendency thus for flow to be dominated by the contributions near the percolation threshold makes the tortuosity of random percolation relevant for dispersion. We calculate the distribution of controlling conductance values, g, using cluster statistics of percolation theory. We relate this to a distribution of arrival times, t, by relating t to g. We find the time of transit t along a straight path through the appropriate distribution of pore sizes and then modify using an expression for the tortuosity of the percolation backbone cluster. We find that under certain conditions the distribution of arrival times has a long tail (infinite variance), but under other conditions the variance is finite. We hope to relate these results to relevant experiments.
U43B-1125
Percolating Magmas in Three Dimensions
The classical models of volcanic eruptions assume that they originate in the magma as a consequence of critical stresses or critical strain rates being exceeded followed by catastrophic fragmentation of the magma. 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 called the percolation threshold, even in the absence of external stresses the magma fragments. By considering 2D percolation with the (observed) extreme power law bubble distributions, we showed numerically that P2c had the apparently realistic value of 0.7. However, the properties of percolating systems are significantly different in 2D and 3D. We will 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 3D critical percolation points, b) a generic result of 3D percolation is that the resulting primary fragments will have power law distributions with exponent B3f of 1.186±0.002. We will review the relevant percolation literature and point out that the elastic properties may have lower – possibly more realistic - critical vesicularities relevant to magmas. We will then explore the implication of long-range correlations and discuss this in combination with bubble anisotropy.
U43B-1126
Flow Dimension and Anomalous Diffusion Coefficient as Potential Indicators of the Connectivity of a Fractured Aquifer
The General Radial Flow (GRF) interpretation of hydraulic tests presented by Barker (1988) has been increasingly applied in recent years due to its capability to address the complex geometry of fractured rock aquifers. This approach considers an additional parameter, namely the flow dimension n, to describe the flow geometry. While the published literature has addressed the analysis and characterization of connectivity in a fracture network system (Bour and Davy, 1997), few studies have related fracture connectivity to the behavior of pressure transients and the flow dimension. Studies relating the flow dimension to transport processes are scarcer still. The present work analyzes the connectivity of a fracture network system versus the behavior of the flow dimension and the anomalous diffusion coefficient. A discrete linear feature network with lengths distributed as a power-law is used to represent the connectivity of a fractured rock aquifer. An intensive Monte Carlo simulation of transient hydraulic tests then examines the effect of the connectivity on the behavior of the flow dimension and the diffusivity of the system at different spatial scales of the aquifer test. Preliminary results show that the behavior of the flow dimension and the diffusivity coefficient depend on the fracture length distribution, hence on the connectivity regime of the fracture network system. While the flow dimension stabilizes to a value less than the Euclidean dimension at late time of the aquifer test and for a particular connectivity regime, the diffusivity coefficient confirms the non-fickean behavior of the transport processes. Therefore, both the flow dimension and diffusivity coefficient could be used as potential indicators to identify connectivity regimes that better represent the observed behavior of flow and transport in fracture media.
U43B-1127
Characteristic timescales of hydrologic parameters in Gwangneung forest catchment, Korea
In the Gwangneung forest, a Korean National Arboretum, fluxes of energy, water and CO2 have been monitored continuously over the forest catchment as a part of HydroKorea project. Using variograms, characteristic time scales of hydrologic parameters including precipitation, stream discharge, soil moisture and groundwater levels in response to the rainfall were evaluated to be less than 6 hours, 5 to 10 hours, less than 10 hours, and greater than a day, respectively. The range of timescale of each parameter could be attributed to the variation of corresponding rainfall events. In addition, the results show that there are temporal correlations among hydrologic parameters and the filtering effect indicating the small time-scale variation is gradually removed as precipitation moves downward to the deeper ground. http://koflux.yonsei.ac.kr /carbo_hydrokorea.htm
U43B-1128
Multifractal Flood Frequency Analysis
Hydrology and more generally sciences involved in water resources management, researches and technological or operational development face a fundamental difficulty: the extreme variability of hydrological fields. It clearly appears today that this variability is a function of the observation scale and yield natural hazards such as floods or droughts. The estimation of return periods for extreme precipitation and flooding events requires a model of the natural (unperturbed) statistical behaviour of the probability tails and the possible clustering (including possible long-range dependencies) of the extremes. Appropriate approaches for handling such non classical variability over wide ranges of time and space scale do exist. They are based on a fundamental property of the non-linear equations: scale invariance. Its specific framework is that of multifractals. In this framework hydrological variability builds up scale by scale leading to non-classical statistics; this provides the key element needed to better understand and predict floods. Scaling is a verifiable physical principle which can be exploited to model hydrological processes and estimate their statistics over wide ranges of space-time scales. We first present the Multifractal Flood Frequency Analysis (MFFA) tool and illustrate some results of its application to a large database (for more than 16000 selected stations over USA and Canada). We then discuss its efficiency by showing how the mean flow information – coupled with universal multifractal parametrizations with power law tails – can be used to estimate return times for extreme flood events.
U43B-1129
Channel Networks Driven by Subsurface Flow Break Scale Invariance
Broadly viewed, there are two qualitatively different mechanisms by which flowing water shapes topography. In the more common case, flow over land exerts a shear stress that erodes material. In the second case, erosion occurs when groundwater emerges at the base of channel heads. Both mechanisms---called, respectively, erosion by overland flow and erosion by subsurface, seepage flows---typically lead to the growth of ramified channel networks. In overland flow, variations in topography focus the flow of water; thus the long term evolution of the network depends sensitively on any initial variations in topography. In contrast, the focusing of subsurface flow is due to variations in the elevation of the water table, which is sensitive to the network geometry but relatively insensitive to small variations in topography. Consequently the growth of seepage-driven networks is in many respects a simpler process that derives more from the local competition of channels for water and less from small, non-local perturbations to topography. Here we report an empirical and theoretical study of networks driven by seepage. We hypothesize that seepage networks evolve via two mechanisms: the "nucleation" of new channels and their growth. Nucleation occurs at a rate proportional to drainage density while channels grow at a velocity proportional to flux through channel heads. We show that geometric features of the resulting network depend on the ratio of the characteristic time scale between the nucleation of new channel heads and the characteristic time scale over which a nucleated channel head advances to the groundwater divide. If new channels nucleate either much faster or much slower than the time required for existing tips to grow to the divide, the resulting dynamics produce networks that are scale invariant. The intermediate case, which necessarily occurs between the largest and smallest length scales, is however scale dependent. We test these ideas by analyzing a high-resolution digital elevation model of a kilometer-scale network of seepage-driven channels located on the Florida Panhandle. For each channel, we compute a dimensionless quantity η that is related to basin size and shape, and show that η varies with length scale as in our model. In contrast, the same measure confirms the scale-invariance of models of networks driven by overland flow. We also report studies of large-scale river networks. The breakdown of scale invariance in seepage networks exhibits a stark contrast to the well-known fractal properties of river networks driven by overland flow. This suggests that a quantitative distinction between seepage and overland flow may be possible in instances, such as Martian drainage networks, where the network's provenance is unclear.
U43B-1130
Evidence for the Dynamical Origin of Ordered Channel Networks Driven by Subsurface Flow
Channel networks driven by subsurface flow exhibit a geometrically ordered structure distinguished in part by a characteristic spacing between channels. Here we ask whether such patterns are determined solely by local inhomogeneities, or if intrinsic growth dynamics likely plays a significant role. We study an especially ramified km-scale section of Little Sweetwater Creek located near Bristol, Florida. The creek is part of a larger network of channels incised in a bed of homogeneous sand, forming a particularly simple system. Erosion occurs primarily at channel heads due to outward seepage of subsurface water. Consequently, knowledge of spatial variation in the water table is sufficient to determine spatial variation in erosive driving. We performed a high-resolution three-dimensional ground-penetrating radar survey to determine the depth of the water table near the creek perimeter. Comparison of water table height with different features of the local environment shows that it does not closely follow either surface elevation or larger-scale topography; however, it does correlate well with the local distance to the nearest point on the channel. Moreover, the height of the water table rises roughly logarithmically with that distance. This correlation, along with a supporting theoretical model and numerical simulations, provides strong evidence in favor of an intrinsic dynamic mechanism for the formation and growth of this ordered network. Such a mechanism implies that regional geometric factors are more important than local inhomogeneities in determining this network structure.
U43B-1131
Isotropic turbulence, stable layers: atmospheric fictions
Using state of the art drop sonde data (from 237 sondes over the Pacific) we examine two classical and fundamental idealizations of atmospheric science showing that they are untenable in the light of the vertical structure. The first is the notion of stable atmospheric layers. This is used for understanding atmospheric dynamics and thermodynamics, including notions such as potential vorticity. Using the drop sonde data, we show that each apparently stable layer is actually composed of a hierarchy of unstable layers themselves with embedded stable sublayers, each with unstable sub-sub layers etc. i.e. in a Russian doll-like fractal hierarchy whose dimension we estimate. We therefore argue that the notion is untenable and must be replaced by modern scaling notions. Although the same basic conclusion follows for conditional, dynamical and convective stability, we showed that their correlation exponent (quantifying their sparsenesses) were 0.36±0.056, 0.22±0.037, 0.15±0.016 respectively. The second idealization we examine is the turbulence assumption of isotropy. If we include intermittency, Kolmorogov's landmark proposal that fully developed turbulence has an "inertial subrange" with isotropic energy spectrum E(k) ≈ k**-β with β≈5/3 has apparently been spectacularly confirmed in both the horizontal direction and in the time domain (k is a wavenumber). For gradients over a horizontal distance Δx this implies Δv≈Δz**Hh (Hh=1/3 corresponds to β=5/3; "<.>" indicates ensemble averaging). Remarkably, Hv for gradients over vertical distances Δz (Δv≈Δz**Hv) has not been seriously investigated. Using drop sonde data of horizontal wind, we find that from scales of 5 m to >10 km from the surface layer through to the top of the troposphere, Hv is close to (or larger) than the Bolgiano-Obukhov value 3/5. Hv>Hh implies that a) the atmosphere becomes progressively less stratified at smaller scales although in a scaling way; b) that at most a single (roughly) isotropic "sphero-scale" exists (often in the range 1- 100 cm).
U43B-1132
Scaling properties of meteorological analyses, numerical models, and atmospheric fields
Between the outer planetary scale and inner (viscous) dissipation scale, the basic equations of the atmosphere have no characteristic lengths. We therefore expect that both the atmosphere and the corresponding numerical (weather/climate) models should be scaling; i.e., that their statistics (such as spectra) are power law functions of space/time scales. This expectation has been repeatedly confirmed by empirical observations, and most recently and spectacularly, by the systematic analysis of TRMM satellite data which include radar-reflectivity, visible, near and far infrared, and passive microwave channels. While the temporal scaling properties of climate models have been occasionally studied, the model spatial resolutions have been too low to allow systematic study of their spatial scaling properties. However, in the last few years, the models have become large enough (i.e., they contain a wide enough range of spatial scales) so that their spatial scaling properties can be reasonably well determined using a variety of analysis techniques. It is therefore possible to evaluate model performance not only in the usual deterministic sense of comparing a model realization and an atmospheric "snapshot", but also by making a stochastic evaluation by comparing their scale-by-scale statistical properties. This overcomes many of the problems of inadequate data which plague attempts to evaluate performance on individual realizations. Indeed, by systematically studying the scaling characteristics of the empirical data, the analyses, and then the model integrations, we can examine the "stochastic coherence" of the data assimilation and model system. We investigate this problem by considering both temporal and spatial scaling in the ERA-40 (ECMWF reanalysis) and CMC GEMS (Canadian Meteorological Centre Global Environmental Multi-Scale) model.
U43B-1133
Beyond multi-fractals: surrogate time series and fields
Most natural complex are characterised by variability on a large range of temporal and spatial scales. The two main methodologies to generate such structures are Fourier/FARIMA based algorithms and multifractal methods. The former is restricted to Gaussian data, whereas the latter requires the structure to be self-similar. This work will present so-called surrogate data as an alternative that works with any (empirical) distribution and power spectrum. The best-known surrogate algorithm is the iterative amplitude adjusted Fourier transform (IAAFT) algorithm. We have studied six different geophysical time series (two clouds, runoff of a small and a large river, temperature and rain) and their surrogates. The power spectra and consequently the 2nd order structure functions were replicated accurately. Even the fourth order structure function was more accurately reproduced by the surrogates as would be possible by a fractal method, because the measured structure deviated too strong from fractal scaling. Only in case of the daily rain sums a fractal method could have been more accurate. Just as Fourier and multifractal methods, the current surrogates are not able to model the asymmetric increment distributions observed for runoff, i.e., they cannot reproduce nonlinear dynamical processes that are asymmetric in time. Furthermore, we have found differences for the structure functions on small scales. Surrogate methods are especially valuable for empirical studies, because the time series and fields that are generated are able to mimic measured variables accurately. Our main application is radiative transfer through structured clouds. Like many geophysical fields, clouds can only be sampled sparsely, e.g. with in-situ airborne instruments. However, for radiative transfer calculations we need full 3-dimensional cloud fields. A first study relating the measured properties of the cloud droplets and the radiative properties of the cloud field by generating surrogate cloud fields yielded good results within the measurement error. A further test of the suitability of the surrogate clouds for radiative transfer is evaluated by comparing the radiative properties of model cloud fields of sparse cumulus and stratocumulus with their surrogate fields. The bias and root mean square error in various radiative properties is small and the deviations in the radiances and irradiances are not statistically significant, i.e. these deviations can be attributed to the Monte Carlo noise of the radiative transfer calculations. We compared these results with optical properties of synthetic clouds that have either the correct distribution (but no spatial correlations) or the correct power spectrum (but a Gaussian distribution). These clouds did show statistical significant deviations. For more information see: http://www.meteo.uni-bonn.de/venema/themes/surrogates/
U43B-1134
Mutltifractal Predictability and Mutltifractal Forecasts
Multifractals have been more and more recognized as powerful tools to analyze spatial heterogeneities and/or temporal variability of complex fields. They are indeed extremely powerful to analyze together space and time fluctuations, in particular their scaling anisotropy. We argue that multifractals are not limited to analyze: we show that they can be further exploited to first study the predictability of complex space-time fields and secondly to obtain optimal forecasts. The intrinsic predictability limits of space time scaling systems, e.g. dynamics coupled with various fields such as the water content, are quite different from those of systems that are only complex in time. Indeed, space time scaling systems do not yield characteristic times of predictability: a limited uncertainty on initial and/or boundary conditions on a given range of time and space scales rapidly grows across the scales and yields power-law decays of the predictability instead of exponential decays. Furthermore, the predictability decay is highly intermittent: the loss of information occurs by intermittent puffs. The predictability itself is multifractal: an infinite hierarchy of power-law exponents is required to characterize the predictability decay from average to extreme events. In particular, we will discuss the multifractal behavior of the decorrelation and correlation fluxes, as well as their required extension. Secondly, this framework allows to proceed to multifractal forecasts in a dynamical manner This can be achieved with the help of ensemble stochastic forecasts, i.e. simulating a given number of possible future realizations and comparing their relative dispersion. We also discuss the possibility to directly achieve multifractal probabilistic forecasts. In both cases, these forecasts are optimal in the sense that they only limited by the intrinsic predictability limits discussed above. http://www.enpc.fr/multifractal/
U43B-1135
Distinguishing Two-Dimensional from Three-Dimensional Turbulence In the Near-Earth Atmosphere
As is well known, atmospheric turbulence effects follow a fractal -5/3rds power law for fluctuations in the inertial subrange cascade region of the spectrum. Many meteorological models invoke closure techniques to simulate the dissipation of energy at scales smaller than are resolvable at the model cell size. However these closure methods generally assume fully developed (three-dimensional) turbulence. However, recent analysis of data suggests that these closure techniques should consider both 2-D and 3-D turbulence influences. It is possible to assess the relative degree of 2-D and 3-D turbulence for a data set in a given time window by evaluating a parameter that assesses the invariants of the turbulent kinetic energy stress tensor. Through a multi-scale analysis, a length scale can be evaluated that implies a cutoff boundary scale between 2-D and 3-D turbulence domains. Diurnal data sets of near-surface conditions analyzed for this parameter indicate a distinctive daily pattern commencing at dawn with a significant increase in this length scale, indicative of strong morning mixing. This is followed by a decreasing scale pattern throughout the day until the following morning. Could this pattern be used in modeling of surface coupling, and are there propagation problems involving angle-of-arrival fluctuations that are influenced by 2-D turbulence?
U43B-1136
Spontaneously Emergent Marginally Stable Highly Non-normal Jets in Turbulent Flows
A remarkable phenomenon in turbulent flows is the spontaneous emergence of coherent large spatial scale jets. Familiar geophysical examples of this phenomenon include the Jovian banded winds and the Earth's polar front jet. These jets are maintained in a nonlinear equilibrium with the much smaller spatial and temporal scale turbulence with which they coexist. Frequently these jets are found to exist in marginally stable states that support large transient growth. In this talk a comprehensive theory for the interaction of jets with turbulence, stochastic structural stability theory (hereafter SSST), is introduced and applied to the problem of understanding the formation and maintenance in turbulent flows of marginally stable but highly non-normal equilibrium jets.