Efficient calculation of the green function for acoustic propagation above a homogeneous impedance plane

This paper is concerned with the problem of propagation from a monofrequency coherent line source above a plane of homogeneous surface impedance. The solution of this problem occurs in the kernel of certain boundary integral equation formulations of acoustic propagation above an impedance boundary, and the discussion of the paper is motivated by this application. The paper starts by deriving representations, as Laplace-type integrals, of the solution and its first partial derivatives. The evaluation of these integral representations by Gauss-Laguerre quadrature is discussed, and theoretical bounds on the truncation error are obtained. Specific approximations are proposed which are shown to be accurate except in the very near field, for all angles of incidence and a wide range of values of surface impedance. The paper finishes with derivations of partial results and analogous Laplace-type integral representations for the case of a point source.

Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere

It is shown that, for a sufficiently large value of β, two-dimensional flow on a doubly-periodic beta-plane cannot be ergodic (phase-space filling) on the phase-space surface of constant energy and enstrophy. A corresponding result holds for flow on the surface of a rotating sphere, for a sufficiently rapid rotation rate Ω. This implies that the higher-order, non-quadratic invariants are exerting a significant influence on the statistical evolution of the flow. The proof relies on the existence of a finite-amplitude Liapunov stability theorem for zonally symmetric basic states with a non-vanishing absolute-vorticity gradient. When the domain size is much larger than the size of a typical eddy, then a sufficient condition for non-ergodicity is that the wave steepness ε < 1, where ε = 2[surd radical]2Z/βU in the planar case and $\epsilon = 2^{\frac{1}{4}} a^{\frac{5}{2}}Z^{\frac{7}{4}}/\Omega U^{\frac{5}{2}}$ in the spherical case, and where Z is the enstrophy, U the r.m.s. velocity, and a the radius of the sphere. This result may help to explain why numerical simulations of unforced beta-plane turbulence (in which ε decreases in time) seem to evolve into a non-ergodic regime at large scales.

Time development of small disturbances to plane Couette Flow

The question of linear sheared-disturbance evolution in constant-shear parallel flow is here reexamined with regard to the temporary-amplification phenomenon noted first by Orr in 1907. The results apply directly to Rossby waves on a beta-plane, and are also relevant to the Eady model of baroclinic instability. It is shown that an isotropic initial distribution of standing waves maintains a constant energy level throughout the shearing process, the amplification of some waves being precisely balanced by the decay of the others. An expression is obtained for the energy of a distribution of disturbances whose wavevectors lie within a given angular wedge and an upper bound derived. It is concluded that the case for ubiquitous amplification made in recent studies may have been somewhat overstated: while carefully-chosen individual Fourier components can amplify considerably before they decay. a general distribution will tend to exhibit little or no amplification.

Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.

Plastic and evolutionary responses to heat stress in a temperate dung fly: negative correlation between basal and induced heat tolerance?

Extreme weather events such as heat waves are becoming more frequent and intense. Populations can cope with elevated heat stress by evolving higher basal heat tolerance (evolutionary response) and/or stronger induced heat tolerance (plastic response). However, there is ongoing debate about whether basal and induced heat tolerance are negatively correlated and whether adaptive potential in heat tolerance is sufficient under ongoing climate warming. To evaluate the evolutionary potential of basal and induced heat tolerance, we performed experimental evolution on a temperate source 4 population of the dung fly Sepsis punctum. Offspring of flies adapted to three thermal selection regimes (Hot, Cold and Reference) were subjected to acute heat stress after having been exposed to either a hot-acclimation or non-acclimation pretreatment. As different traits may respond differently to temperature stress, several physiological and life history traits were assessed. Condition dependence of the response was evaluated by exposing juveniles to different levels of developmental (food restriction/rearing density) stress. Heat knockdown times were highest, whereas acclimation effects were lowest in the Hot selection regime, indicating a negative association between basal and induced heat tolerance. However, survival, adult longevity, fecundity and fertility did not show such a pattern. Acclimation had positive effects in heat-shocked flies, but in the absence of heat stress hot-acclimated flies had reduced life spans relative to nonacclimated ones, thereby revealing a potential cost of acclimation. Moreover, body size positively affected heat tolerance and unstressed individuals were less prone to heat stress than stressed flies, offering support for energetic costs associated with heat tolerance. Overall, our results indicate that heat tolerance of temperate insects can evolve under rising temperatures, but this response could be limited by a negative relationship between basal and induced thermotolerance, and may involve some but not other fitness-related traits.