Existence and homogenization of the Rayleigh-B\'enard problem

The Navier-Stokes equation driven by heat conduction is studied. As a prototype we consider Rayleigh-B\'enard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-B\'enard experiments with Prandtl number close to one, we prove the existence of a global strong solution to the 3D Navier-Stokes equation coupled with a heat equation, and the existence of a maximal B-attractor. A rigorous two-scale limit is obtained by homogenization theory. The mean velocity field is obtained by averaging the two-scale limit over the unit torus in the local variable

Integrating the geodesic equations in the Schwarzschild and Kerr space-times using Beltrami's "geometrical" method

We revisit a little known theorem due to Beltrami, through which the integration of the geodesic equations of a curved manifold is accomplished by a method which, even if inspired by the Hamilton-Jacobi method, is purely geometric. The application of this theorem to the Schwarzschild and Kerr metrics leads straightforwardly to the general solution of their geodesic equations. This way of dealing with the problem is, in our opinion, very much in keeping with the geometric spirit of general relativity. In fact, thanks to this theorem we can integrate the geodesic equations by a geometrical method and then verify that the classical conservation laws follow from these equations.Comment: 12 pages; corrected typos, journal-ref adde

Mapping the ν⊙\nu_\odot Secular Resonance for Retrograde Irregular Satellites

Constructing dynamical maps from the filtered output of numerical integrations, we analyze the structure of the $\nu_\odot$ secular resonance for fictitious irregular satellites in retrograde orbits. This commensurability is associated to the secular angle $\theta = \varpi - \varpi_\odot$, where $\varpi$ is the longitude of pericenter of the satellite and $\varpi_\odot$ corresponds to the (fixed) planetocentric orbit of the Sun. Our study is performed in the restricted three-body problem, where the satellites are considered as massless particles around a massive planet and perturbed by the Sun. Depending on the initial conditions, the resonance presents a diversity of possible resonant modes, including librations of $\theta$ around zero (as found for Sinope and Pasiphae) or 180 degrees, as well as asymmetric librations (e.g. Narvi). Symmetric modes are present in all giant planets, although each regime appears restricted to certain values of the satellite inclination. Asymmetric solutions, on the other hand, seem absent around Neptune due to its almost circular heliocentric orbit. Simulating the effects of a smooth orbital migration on the satellite, we find that the resonance lock is preserved as long as the induced change in semimajor axis is much slower compared to the period of the resonant angle (adiabatic limit). However, the librational mode may vary during the process, switching between symmetric and asymmetric oscillations. Finally, we present a simple scaling transformation that allows to estimate the resonant structure around any giant planet from the results calculated around a single primary mass.Comment: 11 pages, 13 figure

Exact solutions for a class of integrable Henon-Heiles-type systems

We study the exact solutions of a class of integrable Henon-Heiles-type systems (according to the analysis of Bountis et al. (1982)). These solutions are expressed in terms of two-dimensional Kleinian functions. Special periodic solutions are expressed in terms of the well-known Weierstrass function. We extend some of our results to a generalized Henon-Heiles-type system with n+1 degrees of freedom.Comment: RevTeX4-1, 13 pages, Submitted to J. Math. Phy

Breakdown of Lindstedt Expansion for Chaotic Maps

In a previous paper of one of us [Europhys. Lett. 59 (2002), 330--336] the validity of Greene's method for determining the critical constant of the standard map (SM) was questioned on the basis of some numerical findings. Here we come back to that analysis and we provide an interpretation of the numerical results by showing that no contradiction is found with respect to Greene's method. We show that the previous results based on the expansion in Lindstedt series do correspond to the transition value but for a different map: the semi-standard map (SSM). Moreover, we study the expansion obtained from the SM and SSM by suppressing the small divisors. The first case turns out to be related to Kepler's equation after a proper transformation of variables. In both cases we give an analytical solution for the radius of convergence, that represents the singularity in the complex plane closest to the origin. Also here, the radius of convergence of the SM's analogue turns out to be lower than the one of the SSM. However, despite the absence of small denominators these two radii are lower than the ones of the true maps for golden mean winding numbers. Finally, the analyticity domain and, in particular, the critical constant for the two maps without small divisors are studied analytically and numerically. The analyticity domain appears to be an perfect circle for the SSM analogue, while it is stretched along the real axis for the SM analogue yielding a critical constant that is larger than its radius of convergence.Comment: 12 pages, 3 figure

The Measure of the Orthogonal Polynomials Related to Fibonacci Chains: The Periodic Case

The spectral measure for the two families of orthogonal polynomial systems related to periodic chains with N-particle elementary unit and nearest neighbour harmonic interaction is computed using two different methods. The interest is in the orthogonal polynomials related to Fibonacci chains in the periodic approximation. The relation of the measure to appropriately defined Green's functions is established.Comment: 19 pages, TeX, 3 scanned figures, uuencoded file, original figures on request, some misprints corrected, tbp: J. Phys.

Variable δ15N Diet-Tissue Discrimination Factors among Sharks: Implications for Trophic Position, Diet and Food Web Models

The application of stable isotopes to characterize the complexities of a species foraging behavior and trophic relationships is dependent on assumptions of δ(15)N diet-tissue discrimination factors (∆(15)N). As ∆(15)N values have been experimentally shown to vary amongst consumers, tissues and diet composition, resolving appropriate species-specific ∆(15)N values can be complex. Given the logistical and ethical challenges of controlled feeding experiments for determining ∆(15)N values for large and/or endangered species, our objective was to conduct an assessment of a range of reported ∆(15)N values that can hypothetically serve as surrogates for describing the predator-prey relationships of four shark species that feed on prey from different trophic levels (i.e., different mean δ(15)N dietary values). Overall, the most suitable species-specific ∆(15)N values decreased with increasing dietary-δ(15)N values based on stable isotope Bayesian ellipse overlap estimates of shark and the principal prey functional groups contributing to the diet determined from stomach content analyses. Thus, a single ∆(15)N value was not supported for this speciose group of marine predatory fishes. For example, the ∆(15)N value of 3.7‰ provided the highest percent overlap between prey and predator isotope ellipses for the bonnethead shark (mean diet δ(15)N = 9‰) whereas a ∆(15)N value < 2.3‰ provided the highest percent overlap between prey and predator isotope ellipses for the white shark (mean diet δ(15)N = 15‰). These data corroborate the previously reported inverse ∆(15)N-dietary δ(15)N relationship when both isotope ellipses of principal prey functional groups and the broader identified diet of each species were considered supporting the adoption of different ∆(15)N values that reflect the predators' δ(15)N-dietary value. These findings are critical for refining the application of stable isotope modeling approaches as inferences regarding a species' ecological role in their community will be influenced with consequences for conservation and management actions

Variable δ15N Diet-Tissue Discrimination Factors among Sharks: Implications for Trophic Position, Diet and Food Web Models

The application of stable isotopes to characterize the complexities of a species foraging behavior and trophic relationships is dependent on assumptions of δ15N diet-tissue discrimination factors (∆15N). As ∆15N values have been experimentally shown to vary amongst consumers, tissues and diet composition, resolving appropriate speciesspecific ∆15N values can be complex. Given the logistical and ethical challenges of controlled feeding experiments for determining ∆ 15N values for large and/or endangered species, our objective was to conduct an assessment of a range of reported ∆ 15N values that can hypothetically serve as surrogates for describing the predator-prey relationships of four shark species that feed on prey from different trophic levels (i.e., different mean δ 15N dietary values). Overall, the most suitable species-specific ∆ 15N values decreased with increasing dietary-δ 15N values based on stable isotope Bayesian ellipse overlap estimates of shark and the principal prey functional groups contributing to the diet determined from stomach content analyses. Thus, a single ∆ 15N value was not supported for this speciose group of marine predatory fishes. For example, the ∆ 15N value of 3.7‰ provided the highest percent overlap between prey and predator isotope ellipses for the bonnethead shark (mean diet δ 15N = 9‰) whereas a ∆ 15N value \u3c 2.3‰ provided the highest percent overlap between prey and predator isotope ellipses for the white shark (mean diet δ 15N = 15‰). These data corroborate the previously reported inverse ∆ 15N-dietary δ 15N relationship when both isotope ellipses of principal prey functional groups and the broader identified diet of each species were considered supporting the adoption of different ∆ 15N values that reflect the predators’ δ 15N-dietary value. These findings are critical for refining the application of stable isotope modeling approaches as inferences regarding a species’ ecological role in their community will be influenced with consequences for conservation and management actions

The Behavioural and Genetic Mating System of the Sand Tiger Shark, Carcharias taurus, an Intrauterine Cannibal

Sand tiger sharks (Carcharias taurus) have an unusual mode of reproduction, whereby the first embryos in each of the paired uteri to reach a certain size (‘hatchlings’) consume all of their smaller siblings during gestation (‘embryonic cannibalism’ or EC). If females commonly mate with multiple males (‘behavioural polyandry’) then litters could initially have multiple sires. It is possible, however, that EC could exclude of all but one of these sires from producing offspring thus influencing the species genetic mating system (‘genetic monogamy’). Here, we use microsatellite DNA profiling of mothers and their litters (n = 15, from two to nine embryos per litter) to quantify the frequency of behavioural and genetic polyandry in this system. We conservatively estimate that nine of the females we examined (60%) were behaviourally polyandrous. The genetic mating system was characterized by assessing sibling relationships between hatchlings and revealed only 40 per cent genetic polyandry (i.e. hatchlings were full siblings in 60% of litters). The discrepancy stemmed from three females that were initially fertilized by multiple males but only produced hatchlings with one of them. This reveals that males can be excluded even after fertilizing ova and that some instances of genetic monogamy in this population arise from the reduction in litter size by EC. More research is needed on how cryptic post-copulatory and post-zygotic processes contribute to determining paternity and bridging the behavioural and genetic mating systems of viviparous species