H. Gaonac'h

A scaling growth model for bubbles in basaltic lava flows

Abstract Pahoehoe, an and massive lavas from Mount Etna show common statistical properties from one sample to another which are independent of scale/size over certain ranges. The gas vesicle distribution shows two scale-invariant regimes with number density n(V) α V −B−1 where V is the volume and empirically B ≈ 0 for small bubbles and B ≈ 1 for medium to large bubbles. We introduce a bubble growth model which explains the B > 1 range by a strongly non-linear cascading growth regime dominated by a quasi-steady-state coalescence process. The small bubble region is dominated by diffusion; its role is to supply small bubbles to the coalescence regime. The presence of measured dissolved gas in the matrix glass is consistent with the notion that bubbles generally grow in quasi-steady-state conditions. The basic model assumptions are quite robust with respect to the action of a wide variety of processes, since we only require that the dynamics are scaled over the relevant range of scales, and that during the coalescence process, bubble volumes are (approximately) conserved. The model also predicts a decaying coalescence regime (with B > 1) associated with a depletion of the gas source or, alternatively, a loss of large vesicles through the surface of the flow. Our model thus explains the empirical evidence pointing to the coexistence of two different growth mechanisms in subsurface lava flows, but acting over distinct ranges of scale, with non-linear coalescence as the primary growth process. The total vesicularity of each sample can then be well estimated from the partial vesicularity of each growth regime without any outlier problems.

Multifractals and resolution-independent remote sensing algorithms: The example of ocean colour

We argue that geophysical and geographical fields are generally characterised by wide range scaling implying systematic, strong (power law) resolution dependencies when they are remotely sensed. The corresponding geometric structures are fractal sets; the corresponding fields are multifractals. Mathematically, multifractals are measures that are singular with respect to the standard Lebesgue measures; therefore, they are outside the scope of many of the methods of classical geostatistics. Because the resolution of a measurement is generally (due to technical constraints) much larger than the inner scale of the variability/scaling, the observations will be fundamentally observer dependent; hence, standard remote sensing algorithms that do not explicitly take this dependence into account will depend on subjective resolution-dependent parameters. We argue that, on the contrary, the resolution dependence must be systematically removed so that scale-invariant algorithms independent of the observer can be produced. We illustrate these ideas in various ways with the help of eight-channel, 7 m resolution remote ocean colour data (from the MIES II sensor) over the St Lawrence estuary. First, we show that the data is indeed multiscaling over nearly four orders of magnitude in scale, and we quantify this using universal multifractal parameters. With the help of conditional multifractal statistics, we then show how to use multifractals in various practical ways such as for extrapolating from one resolution to another or from one location to another, or to correcting biases introduced when studying extreme, rare phenomena. We also show how the scaling interrelationship of surrogate and in situ data can be handled using vector multifractals and examine the resolution dependence of principle components in dual wavelength analyses. Finally, we indicate why the standard ocean colour algorithms have hidden resolution dependencies, and we show how they can (at least in principle) be removed.

Percolating magmas and explosive volcanism

[1] Magma under pressure rises in conduits, depressurizes, forms bubbles by the exsolution of gas and – at void fractions (P) typically of the order of 0.7 – can fragment and explode. The study of overlapping geometrical units – percolation theory – predicts that at a critical volume fraction Pc the size of the largest simply connected region becomes infinite. We apply percolation theory to overlapping bubbles arguing that this geometric singularity at Pc implies a physical singularity in the magma rheology. This would imply that if the magma is under stress, - whether it is ductile or brittle - this rapid development of a network of infinitely long ‘‘bubbles’’ triggers fragmentation and explosion. Classical monodisperse (equal size) continuum percolation theory predicts Pc = 0.2985 ± 0.005 which is far from the observed values. However, it has recently been shown that the bubble distribution is a power law associated with a huge range of bubble sizes. Using Monte Carlo percolation simulations, we show that distributions exhibiting the empirical exponents are very efficient at ‘‘packing’’ the bubbles, drastically raising Pc to the value = 0.70 ± 0.05. Explosive volcanism is thus explained by singular rheology at Pc. INDEX TERMS: 3220 Mathematical Geophysics: Nonlinear dynamics; 8414 Volcanology: Eruption mechanisms; 8429 Volcanology: Lava rheology and morphology. Citation: Gaonac’h, H., S. Lovejoy, and D. Schertzer, Percolating magmas and explosive volcanism, Geophys. Res. Lett., 30(11), 1559, doi:10.1029/2002GL016022, 2003.

Nonlinear Geophysics: Why We Need It

Few geoscientists would deny that effects are often sensitively dependent on causes, or that their amplification is commonly so strong as to give rise to qualitatively new “emergent” properties, or that geostructures are typically embedded one within another in a hierarchy. Starting in the 1980s, a growing number felt the need to underline the absolute importance of such nonlinearity through workshops and conferences. Building on this, the European Geosciences Union (EGU) organized a nonlinear processes (NP) section in 1990; AGU established a nonlinear geophysics (NG) focus group in 1997; and both unions began collaborating on an academic journal, Nonlinear Processes in Geophysics, in 1994.

Scaling of differentially eroded surfaces in the drainage network of the Ethiopian Plateau

Differentially eroded regions selected over the Ethiopian Plateau, Northeast Africa, were statistically analyzed using satellite images of various electromagnetic spectrum regions and resolutions (Landsat TM and ERS-1). The power spectrum exponent β values for the Landsat TM2 (visible) and ERS-1 images over the same surface are associated to the intrinsic properties of the different sensor type and postprocessing of the data. Differences in the β values were observed between eroded area (Blue Nile Canyon, BNC) and relatively noneroded area (plateau, PLA) for all data sets. These differences are associated to the mechanical erosion of the plateau. The remotely sensed data fields show scaling from 35 m to 15 km, with no break at 1.5 km, and are highly multifractal. Analyses of the Landsat TM bands over each area demonstrated something particular: β values for bands in the shortwave infrared (SWIR) range differed from β values for bands in the visible spectrum range in the plateau area by about 0.48, while in the drainage area, this difference is around 0.13. Landsat TM SWIR bands are sensitive to spectral signature of clay minerals, while data in the visible spectrum range mostly depict topography gradients. Two concurrent processes are highlighted, mechanical erosion and chemical erosion/deposition, which interact to produce the observed differences. In the drainage area, where cliffs and steep slopes are present and mechanical erosion intensively occurs, the alteration minerals are remobilized quickly, whereas in the plateau area, mechanical erosion is low, and alteration mineral deposition is less disturbed. Such new statistical highlights of topographic versus chemical surfaces will have to be taken into account in landforming models.