Generalized Lyapunov exponents indicators in Hamiltonian dynamics: An application to a double star system

The Lyapunov characteristic numbers (LCNs) which are defined as the mean value of the distribution of the local variations of the tangent vectors to the flow (=ln _k ⁱ ) (see Froeschlé, 1984) have been found to be sensitive indicators of stochasticity. So we computed the distribution of these local variations and determined the moments of higher order for the integrable and stochastic regions in a binary star system with =0.5.     

The role of bathymetry and directional wave conditions on observed crescentic bar dynamics

Nearshore sandbars are important features in the surf zone of many beaches because they strongly influence the mean circulation and evolving morphology. Due to variations in wave conditions, sandbars can experience cross-shore migration and vary in shape from alongshore uniform (shore-parallel) to alongshore rhythmic (crescentic). Sandbar dynamics have been studied extensively, but existing observational studies usually do not quantify the processes leading to crescentic bar formation and straightening. This study analyses the dynamics of crescentic bar events at the fetch-limited beach of Castelldefels (northwestern Mediterranean Sea, Spain) using 7.5 years of hourly time-exposure video images and detailed wave conditions. The results show that, despite the generally calm wave conditions, the sandbars were very dynamic in the cross-shore and longshore directions. They often migrated rapidly offshore during storms (up to 70 m in one day) and more slowly onshore during post-storm conditions. Crescentic bars were often present at the study site (48% of the time), but only when the sandbar was at least 10 m from the shoreline. They displayed a large variability in wavelengths (100–700 m), alongshore migration speeds (0–50 m/day) and cross-shore amplitudes (5–20 m). Wavelengths increased for larger bar–shoreline distances and the alongshore migration speeds were strongly correlated with the alongshore component of the radiation stresses. Crescentic patterns typically developed during low–medium energetic waves with limited obliquity ( $\theta \lesssim 20$° at 10 m depth), while bar straightening occurred during medium–high energetic waves with strong oblique angles of incidence ( $\theta \gtrsim 15$°). Overall, this study provides further proof for the important role of wave direction in crescentic bar dynamics and highlights the strong dependence of crescentic bar development on the initial bathymetric configuration.     

Hamiltonian perturbation theory on a manifold

This paper deals with Hamiltonian perturbation theory for systems which, like Euler-Poinsot (the rigid body with a fixed point and no torques), are degenerate and do not possess a global system of action-angle coordinates. It turns out that the usual methods of perturbation theory, which are essentially local being based on the construction of normal forms within the domain of a local coordinate system, are not immediately usable to study perturbations of these systems, since degeneracy makes impossible to control that the system does not fall into a singularity of the coordinates. To overcome this difficulty, we develop a global formulation of Hamiltonian perturbation theory, in which the normal forms are globally defined on the phase space manifold. The key for this study lies in the geometry of the fibration by the invariant tori of an integrable degenerate Hamiltonian system, which is described by some generalizations of the Liouville-Arnol'd theorem and is reviewed in the paper. As an application, we provide a global formulation of Nekhoroshev's theorem on the stability for exponentially long times.     

Possible ways to decipher spatial distribution of mineral resources

Regularity of structural patterns can be connected to planetary disjunctive systems (paleosystems). A computerized universal model of these systems should be developed that may be helpful to predict unknown localities of mineral resources controlled by tectonic processes. The specific character of tectonic phenomena is to be respected in geomathematical models. Some suggestions for applications are given.     

On Szebehely's problem for holonomic systems involving generalized potential functions

We consider the problem of finding the generalized potential function V = U _i(q ¹, q ²,..., q ⁿ)q ⁱ + U(q ¹, q ²,...;q ⁿ) compatible with prescribed dynamical trajectories of a holonomic system. We obtain conditions necessary for the existence of solutions to the problem: these can be cast into a system of n – 1 first order nonlinear partial differential equations in the unknown functions U ₁, U ₂,...;, U _n, U. In particular we study dynamical systems with two degrees of freedom. Using adapted coordinates on the configuration manifold M ₂ we obtain, for potential function U(q ¹, q ²), a classic first kind of Abel ordinary differential equation. Moreover, we show that, in special cases of dynamical interest, such an equation can be solved by quadrature. In particular we establish, for ordinary potential functions, a classical formula obtained in different way by Joukowsky for a particle moving on a surface.Work performed with the support of the Gruppo Nazionale di Fisica Matematica (G.N.F.M.) of the Italian National Research Council.     

Precambrian fluvial deposits: Enigmatic palaeohydrological data from the c. 2–1.9 Ga Waterberg Group, South Africa

Precambrian fluvial systems, lacking the influence of rooted vegetation, probably were characterised by flashy surface runoff, low bank stability, broad channels with abundant bedload, and faster rates of channel migration; consequently, a braided fluvial style is generally accepted. Pre-vegetational braided river systems, active under highly variable palaeoclimatic conditions, may have been more widespread than are modern, ephemeral dry-land braided systems. Aeolian deflation of fine fluvial detritus does not appear to have been prevalent. With the onset of large cratons by the Neoarchaean–Palaeoproterozoic, very large, perennial braided river systems became typical. The c. 2.06–1.88 Ga Waterberg Group, preserved within a Main and a smaller Middelburg basin on the Kaapvaal craton, was deposited largely by alluvial/braided-fluvial and subordinate palaeo-desert environments, within fault-bounded, possibly pull-apart type depositories.

Palaeohydrological data obtained from earlier work in the Middelburg basin (Wilgerivier Formation) are compared to such data derived from the correlated Blouberg Formation, situated along the NE margin of the Main basin. Within the preserved Blouberg depository, palaeohydrological parameters estimated from clast size and cross-bed set thickness data, exhibit rational changes in their values, either in a down-palaeocurrent direction, or from inferred basin margin to palaeo-basin centre. In both the Wilgerivier and Blouberg Formations, calculated palaeoslope values (derived from two separate formulae) plot within the gap separating typical alluvial fan gradients from those which characterise rivers (cf. [Blair, T.C., McPherson, J.G., 1994. Alluvial fans and their natural distinction from rivers based on morphology, hydraulic processes, sedimentary processes, and facies assemblages. J. Sediment. Res. A64, 450–489.]). Although it may be argued that such data support possibly unique fluvial styles within the Precambrian, perhaps related to a combination of major global-scale tectono-thermal and atmospheric–palaeoclimatic events, a simpler explanation of these apparently enigmatic palaeoslope values may be pertinent. Of the two possible palaeohydrological formulae for calculating palaeoslope, one provides results close to typical fluvial gradients; the other formula relies on preserved channel-width data. We suggest that the latter will not be reliable due to problematic preservation of original channel-widths within an active braided fluvial system. We thus find no unequivocal support for a unique fluvial style for the Precambrian, beyond that generally accepted for that period and discussed briefly in the first paragraph.     

A Decision Support System for the Evaluation of Eco-engineering Strategies for Slope Protection

A decision support system (DSS) has been developed to assist expert and non-expert users in the evaluation and selection of eco-engineering strategies for slope protection. This DSS combines a qualitative hazard assessment of erosion and mass movements with a detailed catalogue of eco-engineering strategies for slope protection of which the suitability is evaluated in relation to the data entered. The slope decision support system (SDSS) is a knowledge based DSS in which knowledge is stored in frames containing rules that can evaluate the available information for a project, stored as project specific information (PSI) in a data file. The advantages of such a system are that it accepts incomplete information and that the qualitative nature of the information does not instil the user with a sense of unjustified exactitude. By its multidisciplinary and progressive nature, the DSS will be of value during the initial stages of an eco-engineering project when data collection and the potential of different eco-engineering strategies are considered. The accent of the output of the DSS is on the application of eco-engineering strategies for slope protection as an environmentally-friendly solution aiding sustainable development. For its acceptance within the engineering community, the DSS needs to prove its predictive capacity. Therefore, its performance has been benchmarked against successful and unsuccessful cases of slope stabilisation using eco-engineering. The target audience and the areas of application of this DSS are reviewed and the strategies for further development in this area suggested.     

Towards Formulation of a Space-borne System for Early Warning of Floods: Can Cost-Effectiveness Outweigh Prediction Uncertainty?

The three most important components necessary for functioning of an operational flood warning system are: (1) a rainfall measuring system; (2) a soil moisture updating system; and, (3) a surface discharge measuring system. Although surface based networks for these systems can be largely inadequate in many parts of the world, this inadequacy particularly affects the tropics, which are most vulnerable to flooding hazards. Furthermore, the tropical regions comprise developing countries lacking the financial resources for such surface-based monitoring. The heritage of research conducted on evaluating the potential for measuring discharge from space has now morphed into an agenda for a mission dedicated to space-based surface discharge measurements. This mission juxtaposed with two other upcoming space-based missions: (1) for rainfall measurement (Global Precipitation Measurement, GPM), and (2) soil moisture measurement (Hydrosphere State, HYDROS), bears promise for designing a fully space-borne system for early warning of floods. Such a system, if operational, stands to offer tremendous socio-economic benefit to many flood-prone developing nations of the tropical world. However, there are two competing aspects that need careful assessment to justify the viability of such a system: (1) cost-effectiveness due to surface data scarcity; and (2) flood prediction uncertainty due to uncertainty in the remote sensing measurements. This paper presents the flood hazard mitigation opportunities offered by the assimilation of the three proposed space missions within the context of these two competing aspects. The discussion is cast from the perspective of current understanding of the prediction uncertainties associated with space-based flood prediction. A conceptual framework for a fully space-borne system for early-warning of floods is proposed. The need for retrospective validation of such a system on historical data comprising floods and its associated socio-economic impact is stressed. This proposal for a fully space-borne system, if pursued through wide interdisciplinary effort as recommended herein, promises to enhance the utility of the three space missions more than what their individual agenda can be expected to offer.