The interaction of flexural-gravity waves with periodic geometries

A periodic structure of finite extent is embedded within an otherwise uniform two-dimensional system consisting of finite-depth fluid covered by a thin elastic plate. An incident harmonic flexural-gravity wave is scattered by the structure. By using an approximation to the corresponding linearised boundary value problem that is based on a slowly varying structure in conjunction with a transfer matrix formulation, a method is developed that generates the whole solution from that for just one cycle of the structure, providing both computational savings and insight into the scattering process. Numerical results show that variations in the plate produce strong resonances about the ‘Bragg frequencies’ for relatively few periods. We find that certain geometrical variations in the plate generate these resonances above the Bragg value, whereas other geometries produce the resonance below the Bragg value. The familiar resonances due to periodic bed undulations tend to be damped by the plate.

Complex dynamics in simple systems with periodic parameter oscillations

We study systems with periodically oscillating parameters that can give way to complex periodic or nonperiodic orbits. Performing the long time limit, we can define ergodic averages such as Lyapunov exponents, where a negative maximal Lyapunov exponent corresponds to a stable periodic orbit. By this, extremely complicated periodic orbits composed of contracting and expanding phases appear in a natural way. Employing the technique of ϵ-uncertain points, we find that values of the control parameters supporting such periodic motion are densely embedded in a set of values for which the motion is chaotic. When a tiny amount of noise is coupled to the system, dynamics with positive and with negative nontrivial Lyapunov exponents are indistinguishable. We discuss two physical systems, an oscillatory flow inside a duct and a dripping faucet with variable water supply, where such a mechanism seems to be responsible for a complicated alternation of laminar and turbulent phases.

Enrichment and stability: a phenomenological coupling of energy value and carrying capacity

Simple predator–prey models with a prey-dependent functional response predict that enrichment (increased carrying capacity) destabilizes community dynamics: this is the ‘paradox of enrichment’. However, the energy value of prey is very important in this context. The intraspecific chemical composition of prey species determines its energy value as a food for the potential predator. Theoretical and experimental studies establish that variable chemical composition of prey affects the predator–prey dynamics. Recently, experimental and theoretical approaches have been made to incorporate explicitly the stoichiometric heterogeneity of simple predator–prey systems. Following the results of the previous experimental and theoretical advances, in this article we propose a simple phenomenological formulation of the variation of energy value at increased level of carrying capacity. Results of our study demonstrate that coupling the parameters representing the phenomenological energy value and carrying capacity in a realistic way, may avoid destabilization of community dynamics following enrichment. Additionally, under such coupling the producer–grazer system persists for only an intermediate zone of production—a result consistent with recent studies. We suggest that, while addressing the issue of enrichment in a general predator–prey model, the phenomenological relationship that we propose here might be applicable to avoid Rosenzweig’s paradox.

On the quasi-periodic nature of magnetopause flux transfer events

The recurrence rate of flux transfer events (FTEs) observed near the dayside magnetopause is discussed. A survey of magnetopause observations by the ISEE satellites shows that the distribution of the intervals between FTE signatures has a mode value of 3 min, but is highly skewed, having upper and lower decile values of 1.5 min and 18.5 min, respectively. The mean value is found to be 8 min, consistent with previous surveys of magnetopause data. The recurrence of quasi-periodic events in the dayside auroral ionosphere is frequently used as evidence for an association with magnetopause FTEs, and the distribution of their repetition intervals should be matched to that presented here if such an association is to be confirmed. A survey of 1 year's 15-s data on the interplanetary magnetic field (IMF) suggests that the derived distribution could arise from fluctuations in the IMF Bz component, rather than from a natural oscillation frequency of the magnetosphere-ionosphere system.