The stability analysis of a street of Stuart vortices in a rotating frame is performed by integrating the Kelvin–Townsend equations along the mean flow trajectories, using the geometrical optics technique (Lifschitz & Hameiri 1991) for short-wave perturbations. A parallel is drawn between the formulations of this zonal approach and that of rapid distortion theory, better known to the turbulence community. The results presented confirm those obtained by the standard stability analysis based on normal-mode decomposition: depending on the rotation parameter and the oblique mode considered, three unstable zones are identified, related to the centrifugal, elliptic and hyperbolic instabilities, as observed for Taylor–Green cells (Sipp et al. 1999). Anticyclonic rotation is shown to destabilize Stuart vortices through a combination of the elliptical and centrifugal instability mechanisms, depending on the ratio of its rate to the structure core vorticity. Available stability criteria are discussed in the general case of two-dimensional rotating flows, in relation to their streamline topology and the values of the local Rossby number or vorticity.

New solutions of the Boussinesq equations describe the onset of convection as well as the development of collimated bipolar jets near a point source of both heat and gravity. Stability, bifurcation, and asymptotic analyses of these solutions reveal details of jet formation. Convection (with l cells) evolves from the rest state at the Rayleigh number Ra = Racr = (l − 1)l(l + 1)(l + 2). Bipolar (l = 2) flow emerges at Ra = 24 via a transcritical bifurcation: Re = 7(24 − Ra)/(6 + 4Pr), where Re is a convection amplitude (dimensionless velocity on the symmetry axis) and Pr is the Prandtl number. This flow is unstable for small positive values of Re but becomes stable as Re exceeds some threshold value. The high-Re stable flow emerges from the rest state and returns to the rest state via hysteretic transitions with changing Ra. Stable convection attains high speeds for small Pr (typical of electrically conducting media, e.g. in cosmic jets). Convection saturates due to negative ‘feedback’: the flow mixes hot and cold fluids thus decreasing the buoyancy force that drives the flow. This ‘feedback’ weakens with decreasing Pr, resulting in the development of high-speed convection with a collimated jet on the axis. If swirl is imposed on the equatorial plane, the jet velocity decreases. With increasing swirl, the jet becomes annular and then develops flow reversal on the axis. Transforming the stability problem of this strongly non-parallel flow to ordinary differential equations, we find that the jet is stable and derive an amplitude equation governing the hysteretic transitions between steady states. The results obtained are discussed in the context of geophysical and astrophysical flows.

Axisymmetric vortex core flows, in unconfined and confined geometries, are examined using a quasi-one-dimensional analysis. The goal is to provide a simple unified view of the topic which gives insight into the key physical features, and the overall parametric dependence, of the core area evolution due to boundary geometry or far-field pressure variation. The analysis yields conditions under which waves on vortex cores propagate only downstream (supercritical flow) or both upstream and downstream (subcritical flow), delineates the conditions for a Kelvin–Helmholtz instability arising from the difference in core and outer flow axial velocities, and illustrates the basic mechanism for suppression of this instability due to the presence of swirl. Analytic solutions are derived for steady smoothly, varying vortex cores in unconfined geometries with specified far-field pressure and in confined flows with specified bounding area variation. For unconfined vortex cores, a maximum far-field pressure rise exists above which the vortex cannot remain smoothly varying; this coincides with locally critical conditions (axial velocity equal to wave speed) in terms of wave propagation. Comparison with axisymmetric Navier–Stokes simulations and experimental results indicate that this maximum correlates with the appearance of vortex breakdown and marked core area increase in the simulations and experiments. For confined flows, the core stagnation pressure defect relative to the outer flow is found to be the dominant factor in determining conditions for large increases in core size. Comparisons with axisymmetric Navier–Stokes computations show that the analysis captures qualitatively, and in many instances, quantitatively, the evolution of vortex cores in confined geometries. Finally, a strong analogy with quasi-one-dimensional compressible flow is demonstrated by construction of continuous and discontinuous flows as a function of imposed downstream core edge pressure.

This study explores the dynamics of an unstable jet of two-dimensional, incompressible fluid on the beta-plane. In the inviscid limit, standard weakly nonlinear theory fails to give a low-order description of this problem, partly because the simple shape of the unstable normal mode is insufficient to capture the structure of the forming pattern. That pattern takes the form of ‘cat's eyes’ in the vorticity distribution which develop inside the modal critical layers (slender regions to either side of the jet's axis surrounding the levels where the modal wave speed matches the mean flow). Asymptotic expansions furnish a reduced model which is a version of what is known as the single-wave model in plasma physics. The reduced model predicts that the amplitude of the unstable mode saturates at a relatively low level and is not steady. Rather, the amplitude evolves aperiodically about the saturation level, a result with implications for Lagrangian transport theories. The aperiodic amplitude ‘bounces’ are intimately connected with sporadic deformations of the vortices within the cat's eyes. The theory is compared with numerical simulations of the original governing equations. Slightly asymmetrical jets are also studied. In this case the neutral modes along the stability boundary become singular; an extension of the weakly nonlinear theory is presented for these modes.

This article continues an exploration of instabilities of jets in two-dimensional, inviscid fluid on the beta-plane. At onset, for particular choices of the physical parameters, the normal modes responsible for instability have critical levels that coalesce along the axis of the jet. Matched asymptotic expansion (critical layer theory) is used to derive a reduced model describing the dynamics of these modes. Because the velocity profile is locally parabolic in the vicinity of the critical levels the dynamics is richer than in standard critical layer problems. The model captures the inviscid saturation of unstable modes, the excitation of neutral Rossby waves, and the decay of disturbances when there are no discrete normal modes. Inviscid saturation occurs when the vorticity distribution twists up into vortical structures that take the form of either a pair of ‘cat's eye’ patterns straddling the jet axis, or a single row of vortices. The addition of weak viscosity destroys these cat's eye structures and causes the critical layer to spread diffusively. The results are compared with numerical simulations of the governing equations.

A theoretical model for the instability of two-dimensional turbulent boundary layer over compliant surfaces is described. The principal Reynolds stress is modelled by a well-established mixing-length eddy-viscosity formulation of van Driest. The perturbations of the mean velocity and Reynolds stress fields are coupled via the turbulence model. The investigation of instability is carried out from a time-asymptotic spatio-temporal perspective that classifies instabilities as being either convective or absolute. The occurrence of convective and absolute instabilities over viscoelastic compliant layers is elucidated. Compliant surfaces with low damping are susceptible to convective instability, which gives way to an absolute instability when the surfaces become highly damped. The theoretical results are compared against experimental observations of surface waves on elastic and viscoelastic compliant layers.

A large-scale circulation velocity, often called the ‘wind’, has been observed in turbulent convection in the Rayleigh–Bénard apparatus, which is a closed box with a heated bottom wall. The wind survives even when the dynamical parameter, namely the Rayleigh number, is very large. Over a wide range of time scales greater than its characteristic turnover time, the wind velocity exhibits occasional and irregular reversals without a change in magnitude. We study this feature experimentally in an apparatus of aspect ratio unity, in which the highest attainable Rayleigh number is about 1016. A possible physical explanation is attempted.

We use the Leslie–Ericksen theory to simulate the shear flow of tumbling nematic polymers. The objectives are to explore the onset and evolution of the roll-cell instability and to uncover the flow scenario leading to the nucleation of disclinations. With increasing shear rate, four flow regimes are observed: stable simple shear, steady roll cells, oscillating roll cells and irregular patterns with disclinations. In the last regime, roll cells break up into an irregular and uctuating pattern of eddies. The director is swept into the flow direction in formations called ‘ridges’, which under favourable flow conditions split to form pairs of ± 1 disclinations with non-singular cores. The four regimes are generally consistent with experimental observations, but the mechanism for defect nucleation remains to be verified by more detailed measurements.

The influence of drag on the motion of gravity currents over rigid horizontal surfaces is considered analytically using a Chézy model of boundary shear stress. Although the initial motion is governed by a balance between the buoyancy forces and fluid inertia, drag gradually influences the flow. The length and time scales at which these effects become significant are identified. A perturbation series, valid at early times, is constructed to analyse the changes to the velocity and height of the evolving current due to drag. At much later times, a new class of similarity solutions is developed to model the motion which is now governed by a balance between buoyancy and drag. The transition in the dominant forces which govern the dynamics of the flow is examined by numerically integrating the equations of motion for flows generated by a constant flux of relatively dense fluid. The numerical results confirm both the perturbation solution, valid at early times, and the new similarity solution valid at late times. The transition between the two may involve the formation of a discontinuity (bore). Finally particle-driven currents, which exhibit different dynamical behaviour due to the progressive reduction of their density arising from particle sedimentation, are investigated.

A free surface may be deformed by fluid motions; such deformation may lead to surface roughness, breakup, or disintegration. This paper describes the wide range of free-surface deformations that occur when there is turbulence at the surface, and focuses on turbulence in the denser, liquid, medium. This turbulence may be generated at the surface as in breaking water waves, or may reach the surface from other sources such as bed boundary layers or submerged jets. The discussion is structured by consideration of the stabilizing influences of gravity and surface tension against the disrupting effect of the turbulent kinetic energy. This leads to a two-parameter description of the surface behaviour which gives a framework for further experimental and theoretical studies. Much of the discussion is necessarily heuristic, and is often limited by a lack of appropriate experimental observations. It is intended that such experiments be stimulated, to test the value or otherwise of our two-parameter description.

Strong turbulence at a water–air free surface can lead to splashing and a disconnected surface as in a breaking wave. Averaging to obtain boundary conditions for such flows first requires equations of motion for the two-phase region. These are derived using an integral method, then averaged conservation equations for mass and momentum are obtained along with an equation for the turbulent kinetic energy in which extra work terms appear. These extra terms include both the mean pressure and the mean rate of strain and have similarities to those for a compressible fluid. Boundary conditions appropriate for use with averaged equations in the body of the water are obtained by integrating across the two-phase surface layer.

A number of ‘new’ terms arise for which closure expressions must be found for practical use. Our knowledge of the properties of strong turbulence at a free surface is insufficient to make such closures. However, preliminary discussions are given for two simplified cases in order to stimulate further experimental and theoretical studies.

Much of the turbulence in a spilling breaker originates from its foot where turbulent water meets undisturbed water. A discussion of averaging at the foot of a breaker gives parameters that may serve to measure the ‘strength’ of a breaker.

Two time-periodic solutions with genuine three-dimensional structure are numerically discovered for the incompressible Navier–Stokes equation of a constrained plane Couette flow. One solution with strong variation in spatial and temporal structure exhibits a full regeneration cycle, which consists of the formation and breakdown of streamwise vortices and low-velocity streaks; the other one, of gentle variation, represents a spanwise standing-wave motion of low-velocity streaks. These two solutions are unstable and the corresponding periodic orbits in the phase space are connected with each other. A turbulent state wanders around the strong one for most of the time except for occasional escapes from it. As a result, the mean velocity profile and the root-mean-squares of velocity fluctuations of the plane Couette turbulence agree very well with the temporal averages of those of this periodic motion. After an occasional escape from the strong solution, the turbulent state reaches the gentle periodic solution and returns. On the way back, it experiences an overshoot accompanied by strong turbulence activity like an intermittent bursting phenomenon.

The extended mild-slope equation and the modified mild-slope equation have been used successfully to study refraction–diffraction of linear water waves by steep bottom roughness. Their consistency has been questioned. A systematic derivation of these model equations exposes and illuminates their rationale. Their good performance stems from an accurate representation of (Class I) Bragg resonance. As a benchmark test case, we consider scattering by a sloping bottom with random roughness. The rates of scattering found for the mean field in both of the approximate models agree exactly with the full theory for scattering by small roughness. This greatly improves the limited agreement which was found for the mild-slope equation, and establishes the validity of the above model equations. The study involves operator calculus, a powerful method for simplifying problems with variable coefficients. The augmented mild-slope equation serves to consistently derive accurate model equations.

We present a theoretical and numerical investigation of longshore currents driven by breaking waves on beaches, especially barred beaches. The novel feature considered here is that the wave envelope is allowed to vary in the alongshore direction, which leads to the generation of strong dipolar vortex structures where the waves are breaking. The nonlinear evolution of these vortex structures is studied in detail using a simple analytical theory to model the effect of a sloping beach. One of our findings is that the vortex evolution provides a robust mechanism through which the preferred location of the longshore current can move shorewards from the location of wave breaking. Such current dislocation is an often-observed (but ill-understood) phenomenon on real barred beaches.

To underpin our results, we present a comprehensive theoretical description of the relevant wave–mean interaction theory in the context of a shallow-water model for the beach. Therein we link the radiation-stress theory of Longuet-Higgins & Stewart to recently established results concerning the mean vorticity generation due to breaking waves. This leads to detailed results for the entire life-cycle of the mean-flow vortex evolution, from its initial generation by wave breaking until its eventual dissipative decay due to bottom friction.

In order to test and illustrate our theory we also present idealized nonlinear numerical simulations of both waves and vortices using the full shallow-water equations with bottom topography. In these simulations wave breaking occurs through shock formation of the shallow-water waves. We note that because the shallow-water equations also describe the two-dimensional flow of a homentropic perfect gas, our theoretical and numerical results can also be applied to nonlinear acoustics and sound–vortex interactions.

We develop a new approach to numerical modelling of water-wave evolution based on the Zakharov integrodifferential equation and outline its areas of application.

The Zakharov equation is known to follow from the exact equations of potential water waves by the symmetry-preserving truncation at a certain order in wave steepness. This equation, being formulated in terms of nonlinear normal variables, has long been recognized as an indispensable tool for theoretical analysis of surface wave dynamics. However, its potential as the basis for the numerical modelling of wave evolution has not been adequately explored. We partly fill this gap by presenting a new algorithm for the numerical simulation of the evolution of surface waves, based on the Hamiltonian form of the Zakharov equation taking account of quintet interactions. Time integration is performed either by a symplectic scheme, devised as a canonical transformation of a given order on a timestep, or by the conventional Runge–Kutta algorithm. In the latter case, non-conservative effects, small enough to preserve the Hamiltonian structure of the equation to the required order, can be taken into account. The bulky coefficients of the equation are computed only once, by a preprocessing routine, and stored in a convenient way in order to make the subsequent operations vectorized.

The advantages of the present method over conventional numerical models are most apparent when the triplet interactions are not important. Then, due to the removal of non-resonant interactions by means of a canonical transformation, there are incomparably fewer interactions to consider and the integration can be carried out on the slow time scale (O(ε2), where ε is a small parameter characterizing wave slope), leading to a substantial gain in computational efficiency. For instance, a simulation of the long-term evolution of 103 normal modes requires only moderate computational resources; a corresponding simulation in physical space would involve millions of degrees of freedom and much smaller integration timestep.

A number of examples aimed at problems of independent physical interest, where the use of other existing methods would have been difficult or impossible, illustrates various aspects of the implementation of the approach. The specific problems include establishing the range of validity of the deterministic description of water wave evolution, the emergence of sporadic horseshoe patterns on the water surface, and the study of the coupled evolution of a steep wave and low-intensity broad-band noise.

Prompted by the recent experiments of Dietz (1999) on boundary-layer receptivity due to a local roughness interacting with a vortical disturbance in the free stream, this paper undertakes to present a second-order asymptotic theory based on the tripledeck formulation. The asymptotic approach allows us to treat vortical perturbations with a fairly general vertical distribution, and confirms Dietz's conclusion that for the convecting periodic wake in his experiments, the receptivity is independent of its vertical structure and can be fully characterized by its slip velocity at the edge of the boundary layer. As in the case of distributed vortical receptivity, dominant interactions that generate Tollmien–Schlichting waves take place in the upper deck as well as in the so-called edge layer centred at the outer reach of the boundary layer. The initial amplitude of the excited Tollmien–Schlichting wave is determined to O(R−1/8) accuracy, where R is the global Reynolds number. An appropriate superposition formula is derived for the case of multiple roughness elements. A comprehensive comparison is made with Dietz's experimental data, and an excellent quantitative agreement has been found for the first time, thereby resolving some uncertainties about this receptivity mechanism.

A Riemann problem with prescribed initial conditions will produce one of three possible wave patterns corresponding to the propagation of the different discontinuities that will be produced once the system is allowed to relax. In general, when solving the Riemann problem numerically, the determination of the specific wave pattern produced is obtained through some initial guess which can be successively discarded or improved. We here discuss a new procedure, suitable for implementation in an exact Riemann solver in one dimension, which removes the initial ambiguity in the wave pattern. In particular we focus our attention on the relativistic velocity jump between the two initial states and use this to determine, through some analytic conditions, the wave pattern produced by the decay of the initial discontinuity. The exact Riemann problem is then solved by means of calculating the root of a nonlinear equation. Interestingly, in the case of two rarefaction waves, this root can even be found analytically. Our procedure is straightforward to implement numerically and improves the efficiency of numerical codes based on exact Riemann solvers.