A sphere exhibits a prolonged residence time when settling through a stable stratification of miscible fluids due to the deformation of the fluid-density field. Using a Green's function formulation, a first-principles numerically assisted theoretical model for the sphere–fluid coupled dynamics at low Reynolds number is derived. Predictions of the model, which uses no adjustable parameters, are compared with data from an experimental investigation with spheres of varying sizes and densities settling in stratified corn syrup. The velocity of the sphere as well as the deformation of the density field are tracked using time-lapse images, then compared with the theoretical predictions. A settling rate comparison with spheres in dense homogeneous fluid additionally quantifies the effect of the enhanced residence time. Analysis of our theory identifies parametric trends, which are also partially explored in the experiments, further confirming the predictive capability of the theoretical model. The limit of infinite fluid domain is considered, showing evidence that the Stokes paradox of infinite fluid volume dragged by a moving sphere can be regularized by density stratifications. Comparisons with other possible models under a hierarchy of additional simplifying assumptions are also presented.

In this paper, the unsteady evolution of two-dimensional fully nonlinear free-surface gravity–capillary solitary waves is computed numerically in infinite depth. Gravity–capillary wavepacket-type solitary waves were found previously for the full Euler equations, bifurcating from the minimum of the linear dispersion relation. Small and moderate amplitude elevation solitary waves, which were known to be linearly unstable, are shown to evolve into stable depression solitary waves, together with a radiated wave field. Depression waves and certain large amplitude elevation waves were found to be robust to numerical perturbations. Two kinds of collisions are computed: head-on collisions whereby the waves are almost unchanged, and overtaking collisions which are either almost elastic if the wave amplitudes are both large or destroy the smaller wave in the case of a small amplitude wave overtaking a large one.

The paper is concerned with analytical study of inertia-gravity waves in rotating density-stratified ideal fluid confined in a spherical shell. It primarily aims at clarifying the possible role of these motions in deep ocean mixing. Recently, it was found that on the ‘non-traditional’ β-plane inertia-gravity internal waves can propagate polewards beyond their inertial latitude, where the wave frequency equals the local Coriolis parameter, by turning into subinertial modes trapped in the narrowing waveguides around the local minima of buoyancy frequency N. The behaviour of characteristics was established: wave horizontal and vertical scales decrease as the wave advances polewards and tend to zero at a latitude corresponding to an attractor of characteristics. However, the basic questions about wave evolution, its quantitative description and the possibility of its reflection from the critical latitude remain open. The present work addresses these issues by studying the linear inviscid evolution of finite bandwidth wavepackets on the ‘non-traditional’ β-plane past the inertial latitude for generic oceanic stratification. Beyond the inertial latitude, the wave field is confined in narrowing waveguides of three distinct generic types around different local minima of the buoyancy frequency. In the oceanic context, the widest is adjacent to the flat bottom, the thinnest is the upper mixed layer, and the middle one is located between the seasonal and main thermocline. We find explicit asymptotic solutions describing the wave field in the WKB approximation. As a byproduct, the conservation of wave action principle is explicitly formulated for all types of internal waves on the ‘non-traditional’ β-plane. The wave velocities and vertical shear tend to infinity and become singular at the attractor latitude or its vicinity for both monochromatic and finite bandwidth packets. We call this phenomenon singular focusing. These WKB solutions are shown to remain valid up to singularity for the bottom and mid-ocean waveguides. The main conclusion is that even in the inviscid setting the wave evolution towards smaller and smaller horizontal and vertical scales is irreversible: there is no reflection. For situations typical of deep ocean, a simultaneous increase in wave amplitude and decrease of vertical scale causes a sharp increase of vertical shear, which may lead to wave breaking and increased mixing.

Measurements of subsonic air jets show that the peak noise usually occurs when observations are made at small angles to the jet axis. In this paper, we develop further understanding of the mathematical properties of this peak noise by analysing the properties of the overall sound pressure level with an acoustic analogy using isotropy as a paradigm for the turbulence. The analogy is based upon the hyperbolic conservation form of the Euler equations derived by Goldstein (Intl J. Aeroacoust., vol. 1, 2002, p. 1). The mean flow and the turbulence properties are defined by a Reynolds-averaged Navier–Stokes calculation, and we use Green's function based upon a parallel mean flow approximation. Our analysis in this paper shows that the jet noise spectrum can, in fact, be thought of as being composed of two terms, one that is significant at large observation angles and a second term that is especially dominant at small observation angles to the jet axis. This second term can account for the experimentally observed peak jet noise (Lush, J. Fluid Mech., vol. 46, 1971, p. 477) and was first identified by Goldstein (J. Fluid Mech., vol. 70, 1975, p. 595). We discuss the low-frequency asymptotic properties of this second term in order to understand its directional behaviour; we show, for example, that the sound power of this term is proportional to the square of the mean velocity gradient. We also show that this small-angle shear term does not exist if the instantaneous Reynolds stress source strength in the momentum equation itself is assumed to be isotropic for any value of time (as was done previously by Morris & Farrasat, AIAA J., vol. 40, 2002, p. 356). However, it will be significant if the auto-covariance of the Reynolds stress source, when integrated over the vector separation, is taken to be isotropic in all of its tensor suffixes. Although the analysis shows that the sound pressure of this small-angle shear term is sensitive to the statistical properties of the turbulence, this work provides a foundation for a mathematical description of the two-source model of jet noise.