We investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may propagate is broader than with solid-body rotation. Using high-resolution calculations based on a spectral method, we show that, as with solid-body rotation, singular modes with thin shear layers following short-period attractors still exist with differential rotation. They exist even in the case of a full sphere. In the limit of vanishing viscosities, the width of the shear layers seems to weakly depend on the global background shear, showing a scaling in ${E}^{1/ 3} $ with the Ekman number $E$, as in the solid-body rotation case. There also exist modes with thin detached layers of width scaling with ${E}^{1/ 2} $ as Ekman boundary layers. The behaviour of inertial waves with a corotation resonance within the shell is also considered. For cylindrical rotation, waves get dramatically absorbed at corotation. In contrast, for shellular rotation, waves may cross a critical layer without visible absorption, and such modes can be unstable for small enough Ekman numbers.

In gas-liquid flows a certain sudden change of the flow structure may occur, which can be described as a transition from ‘jet flow’ to ‘froth flow’ accompanied by energy dissipation and pressure build-up. Upstream of this phenomenon the gas is the continuous phase; downstream the liquid is the continuous phase. The phenomenon, which has been called ‘mixing shock’, shows some similarity and also some differences with the plane shock wave in gasdynamics. In the first part of this paper the mixing shock is treated as a one-dimensional macroscopic process. With the aid of the laws of conservation of mass, momentum and energy, expressions are obtained for the pressure and entropy change across the mixing process. In addition the stability of the mixing shock in a cylindrical flow channel is treated. Next, a theory that explains the gas entrainment mechanism in the mixing shock is proposed. As an experimental tool a water-air ejector with the water as a driving medium was used. The experiments confirm the macroscopic and the microscopic theory. In the last section of this paper theoretical and experimental evidence is combined to construct a model of the processes that play a role in the shock.

Direct numerical simulations of the velocity and temperature fields for turbulent flow in a channel are used to examine the influence of Prandtl number Pr on turbulent transport. The Reynolds number, based on the half-height of the channel and the friction velocity, is Reτ = 150. Prandtl numbers of 1.0, 0.3, 0.1, 0.05, 0.025 were studied. The bottom and the top walls were kept at constant temperatures of +Tw and −Tw. The influence of Pr on Reynolds transport, on the turbulent diffusivity, ατ, and on the spectral density function of the temperature fluctuations was studied. The observation that spatial variations of the ratio of the turbulent diffusivity to the value observed at Pr = 1.0 are not large is used to propose a method for calculating average temperature fields. The decrease in ατ with decreasing Pr is related to observations of the increased damping of high-wavenumber temperature fluctuations. Molecular conductivity, at smaller Pr, is pictured to act as a filter that renders high-frequency velocity fluctuations ineffective in transporting heat.

Barchan dunes are crescent-shaped formations of sand that can dominate both desert and subaqueous landscapes when the supply of sand is scarce. Because of the complexity and scale of the underlying phenomena, the mechanisms governing the entire process from the genesis to the long-term evolution of barchan fields still remain to be better understood. Herein, we attempt to present a description of this process in a subaqueous environment by employing a large-eddy simulation approach that couples turbulent flow and sand-bed morphodynamics. We show that the seeds of the emergent structure in barchan fields are random turbulent flow motions near the initially flat bed. We also provide high-resolution insights into phenomena such as barchan migration and merging, and show how transverse sand waves are formed and migrate over the barchan horns. Furthermore, the transverse sand waves over the barchan horns are shown to be the seeds of the newly born barchans at the end points of the two horns of a barchan through the process known as calving. To show this, we examine the celerity, wavelength and amplitude of the transverse sand waves over the barchan as they approach the end of its horn. The celerity and wavelength of these transverse sand waves are shown to be the defining factors in determining the frequency of the calving process. The amplitude of the newly born barchans (through calving) is also shown to be associated with the amplitude of the transverse waves near the end of the horn. The simulation data also show that the wavelength of the newly born barchans (the distance between individual dunes) is closely related to that of the transverse sand waves over their maternal barchan. Finally, we use the simulation results to discuss past conclusions derived from theory, conventional models and field observations.

The dependence of the parameters of capillary–gravity ripples on the characteristics of the steep surface waves (in the range 4–20 cm) that excite them is found. For steep 4–6 cm waves calculations are performed on the basis of the improved first Stokes method. Qualitative coincidence of the theoretical results with the experimental data is shown. For 7–20 cm waves the results are obtained by the multiple-scale method where the large-scale motion and the driving force for the ripple are found by the improved first Stokes method. Qualitative agreement between theory and experiment in this wavelength range is achieved.

A comprehensive numerical study on the linear stability of mixed-convection flow in a vertical pipe with constant heat flux is presented with particular emphasis on the instability mechanism and the Prandtl number effect. Three Prandtl numbers representative of different regimes in the Prandtl number spectrum are employed to simulate the stability characteristics of liquid mercury, water and oil. The results suggest that mixed-convection flow in a vertical pipe can become unstable at low Reynolds number and Rayleigh numbers irrespective of the Prandtl number, in contrast to the isothermal case. For water, the calculation predicts critical Rayleigh numbers of 80 and −120 for assisted and opposed flows, which agree very well with experimental values of Rac = 76 and −118 (Scheele & Hanratty 1962). It is found that the first azimuthal mode is always the most unstable, which also agrees with the experimental observation that the unstable pattern is a double spiral flow. Scheele & Hanratty's speculation that the instability in assisted and opposed flows can be attributed to the appearance of inflection points and separation is true only for fluids with O(1) Prandtl number. Our study on the effect of the Prandtl number discloses that it plays an active role in buoyancy-assisted flow and is an indication of the viability of kinematic or thermal disturbances. It profoundly affects the stability of assisted flow and changes the instability mechanism as well. For assisted flow with Prandtl numbers less than 0.3, the thermal–shear instability is dominant. With Prandtl numbers higher than 0.3, the assisted-thermal–buoyant instability becomes responsible. In buoyancy-opposed flow, the effect of the Prandtl number is less significant since the flow is unstably stratified. There are three distinct instability mechanisms at work independent of the Prandtl number. The Rayleigh–Taylor instability is operative when the Reynolds number is extremely low. The opposed-thermal–buoyant instability takes over when the Reynolds number becomes higher. A still higher Reynolds number eventually leads the thermal–shear instability to dominate. While the thermal–buoyant instability is present in both assisted and opposed flows, the mechanism by which it destabilizes the flow is completely different.

The structure of wall pressure fluctuations beneath a turbulent boundary layer interacting with a normal shock wave at Mach number M∞ = 1.3 is studied exploiting a direct numerical simulation database. Upstream of the interaction, in the zero-pressure-gradient region, pressure statistics compare well with canonical low-speed boundary layers in terms of fluctuation intensities, space–time correlations, convection velocities and frequency spectra. Across the interaction zone, the root-mean-square wall pressure fluctuations attain very large values (in excess of 162 dB), with a maximum increase of about 7 dB from the upstream level. The two-point wall pressure correlations become more elongated in the spanwise direction, indicating an increase of the pressure-integral length scales, and the convection velocities (determined from space–time correlations) are reduced. The interaction qualitatively modifies the shape of the frequency spectra, causing enhancement of the low-frequency Fourier modes and inhibition of the higher ones. In the recovery region past the interaction, the pressure spectra collapse very accurately when scaled with either the free-stream dynamic pressure or the maximum Reynolds shear stress, and exhibit distinct power-law regions with exponent −7/3 at intermediate frequencies and −5 at high frequencies. An analysis of the pressure sources in the Lighthill's equation for the instantaneous pressure has been performed to understand their contributions to the wall pressure signature. Upstream of the interaction the sources are mainly located in the proximity of the wall, whereas past the shock, important contributions to low-frequency pressure fluctuations are associated with long-lived eddies developing far from the wall.

A turbulent lifted slot-jet flame is studied using direct numerical simulation (DNS). A one-step chemistry model is employed with a mixture-fraction-dependent activation energy which can reproduce qualitatively the dependence of the laminar burning rate on the equivalence ratio that is typical of hydrocarbon fuels. The basic structure of the flame base is first examined and discussed in the context of earlier experimental studies of lifted flames. Several features previously observed in experiments are noted and clarified. Some other unobserved features are also noted. Comparison with previous DNS modelling of hydrogen flames reveals significant structural differences. The statistics of flow and relative edge-flame propagation velocity components conditioned on the leading edge locations are then examined. The results show that, on average, the streamwise flame propagation and streamwise flow balance, thus demonstrating that edge-flame propagation is the basic stabilisation mechanism. Fluctuations of the edge locations and net edge velocities are, however, significant. It is demonstrated that the edges tend to move in an essentially two-dimensional (2D) elliptical pattern (laterally outwards towards the oxidiser, then upstream, then inwards towards the fuel, then downstream again). It is proposed that this is due to the passage of large eddies, as outlined in Su et al. (Combust. Flame, vol. 144 (3), 2006, pp. 494–512). However, the mechanism is not entirely 2D, and out-of-plane motion is needed to explain how flames escape the high-velocity inner region of the jet. Finally, the time-averaged structure is examined. A budget of terms in the transport equation for the product mass fraction is used to understand the stabilisation from a time-averaged perspective. The result of this analysis is found to be consistent with the instantaneous perspective. The budget reveals a fundamentally 2D structure, involving transport in both the streamwise and transverse directions, as opposed to possible mechanisms involving a dominance of either one direction of transport. It features upstream transport balanced by entrainment into richer conditions, while on the rich side, upstream turbulent transport and entrainment from leaner conditions balance the streamwise convection.

The process of the formation and early evolution of a condensation trail (‘contrail’) in the near field of an aircraft wake was numerically studied by means of a mixed Eulerian/Lagrangian two-phase flow approach. Large-eddy simulations were used for the carrier phase, while, for the dispersed phase, a Lagrangian particle tracking method was used, coupled with a microphysics model to account for ice nucleation. The basic configuration was an exhaust engine jet loaded with soot particles and water vapour and interacting with a wing-tip trailing vortex. The thermodynamic conditions for contrail formation were identified by tracking the spatial distribution of supersaturation around particles. A strong mass coupling between the two phases was demonstrated by the simulations: the condensation of water vapour over soot particles, induced by exhaust dispersion into cold ambient air, leads to the formation of ice crystals whose size grows until thermodynamic equilibrium between the two phases is reached. Finally, local vapour depletion causes significant deviation from the classical mixing line theory and is also responsible for polydispersion of particle radii.

Experiments on the instability of the interface in two-phase displacements in Hele-Shaw cells were conducted using air and a viscous oil as the working fluids. The experiments had two objectives: (i) to provide quantitative measurements of the growth constants of the instability which occurs when a less-viscous fluid displaces a more-viscous one, and (ii) to compare the measured dispersion relations with the predictions of the recent theory of Park & Homsy (1984). The experiments were made by analysing the growth characteristics of between 10 and 20 Fourier modes describing the shape of the interface between displaced and displacing fluids, using still photography. For capillary numbers Ca = μU/γ less than approximately 4 × 10−3 the agreement is only fair, owing to substantial edge effects produced by a nearly static contact line near the lateral boundaries of the cell. For 4 × 10−3 < Ca < 1 × 10−2 theory and experiment agree to within the accuracy of the measurements. Location and verification of the behaviour of modes near the predicted cut-off wavenumber give partial verification of the theory of Park & Homsy.

The valveless impedance pump is a simple design that allows the producion or amplification of a flow without the requirement for valves or impellers. It is based on fluid-filled flexible tubing, connected to tubing of different impedances. Pumping is achieved by a periodic excitation at an off-centre position relative to the tube ends. This paper presents a comprehensive study of the fluid and structural dynamics in an impedance pump model using numerical simulations. An axisymmetric finite-element model of both the fluid and solid domains is used with direct coupling at the interface. By examining a wide range of parameters, the pump's resonance nature is described and the concept of resonance wave pumping is discussed. The main driving mechanism of the flow in the tube is the reflection of waves at the tube boundary and the wave dynamics in the passive tube. This concept is supported by three different analyses: (i) time-dependent pressure and flow wave dynamics along the tube, (ii) calculations of pressure–flow loop areas along the passive tube for a description of energy conversion, and (iii) an integral description of total work done by the pump on the fluid. It is shown that at some frequencies, the energy given to the system by the excitation is converted by the elastic tube to kinetic energy at the tube outlet, resulting in an efficient pumping mechanism and thus significantly higher flow rate. It is also shown that pumping can be achieved with any impedance mismatch at one boundary and that the outlet configuration does not necessarily need to be a tube.

A gas in a tube is excited by a reciprocating piston operating at or near a resonant frequency. Damping is introduced into the system by two means: radiation of energy from one end of the tube and rate dependence of the gas. These define a lumped damping coefficient. It is shown that in the small rate limit the signal in the periodic state suffers negligible distortion in one travel time, and hence its propagation according to acoustic theory is valid. The shape of the signal is determined by a nonlinear ordinary differential equation. The small rate condition provides a test of the applicability of the theory to given experimental conditions. When there is no damping, shocks are a feature of the flow for frequencies in the resonant band. For a given amount of damping an upper bound on the piston acceleration which ensures shockless motion is given. The resonant band is analysed for both damped and undamped cases. The predictions of the theory are compared with experiment.

When a container of fluid of arbitrary shape is heated from below and the temperature gradient exceeds a critical value ([Rscr ]c2) the conduction solution with no motion becomes unstable and is replaced by convection. The convection may have two forms: one with ‘upflow’ at the centre of the container and one with ‘downflow’ there. Here we study the stability of the two forms of convection. Both forms are here shown to be stable to infinitesimal disturbances. When the viscosity varies with the temperature or the conduction profile is not linear, etc., the steady convection can be driven with finite amplitudes |ε| at subcritical values of the temperature contrast ([Rscr ]2 < [Rscr ]c2). This subcritical convection is stable when the convection is strong (|ε| > |ε*| > 0) but is unstable when the convection is feeble (|ε| > |ε*|). Hence, when |ε| > |ε*| and [Rscr ]2 < [Rscr ]c2 either ‘upflow’ or ‘downflow’, but not both, is stable. When [Rscr ]2 > [Rscr ]c2, however, both the ‘upflow’ and the ‘downflow’ can be stable. This contrasts with the corresponding situation which is known to hold when the container is an unbounded layer. In the layer there is only one stable form of convection. The difference between the bounded domain with two forms of convection and the layer with just one stable form is traced to the mathematical property of simplicity of [Rscr ]c2 when viewed as an eigenvalue of the linear stability problem for the conduction solution. It is argued that [Rscr ]c2 is a simple eigenvalue in most domains, but in the layer [Rscr ]c2 can have infinite multiplicity. The explanation of the transition from the bounded domain to the unbounded layer is sought (1) in the chaotic conditions which frequently prevail at the edges of a ‘bounded’ layer and (2) in the fact that in the layer of large horizontal extent, the higher eigenvalues crowd [Rscr ]c2. In the course of the explanation, a new exact solution of the linear Bénard problem in a cylinder with a rigid side wall and a stress-free top and bottom is derived.

Vortex-induced vibration (VIV) of a sphere represents one of the most generic fundamental fluid–structure interaction problems. Since vortex-induced vibration can lead to structural failure, numerous studies have focused on understanding the underlying principles of VIV and its suppression. This paper reports on an experimental investigation of the effect of imposed axial rotation on the dynamics of vortex-induced vibration of a sphere that is free to oscillate in the cross-flow direction, by employing simultaneous displacement and force measurements. The VIV response was investigated over a wide range of reduced velocities (i.e. velocity normalised by the natural frequency of the system): $3\leqslant U^{\ast }\leqslant 18$, corresponding to a Reynolds number range of $5000<Re<30\,000$, while the rotation ratio, defined as the ratio between the sphere surface and inflow speeds, $\unicode[STIX]{x1D6FC}=|\unicode[STIX]{x1D714}|D/(2U)$, was varied in increments over the range of $0\leqslant \unicode[STIX]{x1D6FC}\leqslant 7.5$. It is found that the vibration amplitude exhibits a typical inverted bell-shaped variation with reduced velocity, similar to the classic VIV response for a non-rotating sphere but without the higher reduced velocity response tail. The vibration amplitude decreases monotonically and gradually as the imposed transverse rotation rate is increased up to $\unicode[STIX]{x1D6FC}=6$, beyond which the body vibration is significantly reduced. The synchronisation regime, defined as the reduced velocity range where large vibrations close to the natural frequency are observed, also becomes narrower as $\unicode[STIX]{x1D6FC}$ is increased, with the peak saturation amplitude observed at progressively lower reduced velocities. In addition, for small rotation rates, the peak amplitude decreases almost linearly with $\unicode[STIX]{x1D6FC}$. The imposed rotation not only reduces vibration amplitudes, but also makes the body vibrations less periodic. The frequency spectra revealed the occurrence of a broadband spectrum with an increase in the imposed rotation rate. Recurrence analysis of the structural vibration response demonstrated a transition from periodic to chaotic in a modified recurrence map complementing the appearance of broadband spectra at the onset of bifurcation. Despite considerable changes in flow structure, the vortex phase ($\unicode[STIX]{x1D719}_{vortex}$), defined as the phase between the vortex force and the body displacement, follows the same pattern as for the non-rotating case, with the $\unicode[STIX]{x1D719}_{vortex}$ increasing gradually from low values in Mode I of the sphere vibration to almost $180^{\circ }$ as the system undergoes a continuous transition to Mode II of the sphere vibration at higher reduced velocity. The total phase ($\unicode[STIX]{x1D719}_{total}$), defined as the phase between the transverse lift force and the body displacement, only increases from low values after the peak amplitude response in Mode II has been reached. It reaches its maximum value (${\sim}165^{\circ }$) close to the transition from the Mode II upper plateau to the lower plateau, reminiscent of the behaviour seen for the upper to lower branch transition for cylinder VIV. Hydrogen-bubble visualisations and particle image velocimetry (PIV) performed in the equatorial plane provided further insights into the flow dynamics near the sphere surface. The mean wake is found to be deflected towards the advancing side of the sphere, associated with an increase in the Magnus force. For higher rotation ratios, the near-wake rear recirculation zone is absent and the flow is highly vectored from the retreating side to the advancing side, giving rise to large-scale shedding. For a very high rotation ratio of $\unicode[STIX]{x1D6FC}=6$, for which vibrations are found to be suppressed, a one-sided large-scale shedding pattern is observed, similar to the shear-layer instability one-sided shedding observed previously for a rigidly mounted rotating sphere.

This paper addresses shear-induced self-diffusion in a monodisperse suspension of non-Brownian particles in Couette flow by two-dimensional computer simulations following the lattice-Boltzmann method. This method is suited for the study of (many-particle) particulate suspensions and can not only be applied for Stokes flow, but also for flow with finite Reynolds number. At relatively low shear particle Reynolds numbers (up to 0.023), shear-induced diffusivity exhibited a linear dependence on the shear rate, as expected from theoretical considerations. Simulations at shear particle Reynolds numbers between 0.023 and 0.35, however, revealed that in this regime, shear-induced diffusivity did not show this linear dependence anymore. Instead, the diffusivity was found to increase more than linearly with the shear rate, an effect that was most pronounced at lower area fractions of 0.10 and 0.25. In the same shear regime, major changes were found in the flow trajectories of two interacting particles in shear flow (longer and closer approach) and in the viscosity of the suspension (shear thickening). Moreover, the suspended particles exhibited particle clustering. The increase of shear-induced diffusivity is shown to be directly correlated with this particle clustering. As for shear-induced diffusivity, the effect of increasing shear rates on particle clustering was the most intensive at low area fractions of 0.10 and 0.25, where the radius of the clusters increased from about 4 to about 7 particle radii with an increase of the shear Reynolds number from 0.023 to 0.35. The importance of particle clustering to shear-induced diffusion might also indicate the importance of other factors that can induce particle clustering, such as, for example, colloidal instability.

A theoretical model is reported that describes wave propagation between three floating Euler–Bernoulli thin elastic sheets extending, respectively, from $-\infty$ to 0, from 0 to $l$, and from $l$ to $\infty$, with properties, e.g. thickness, that can be specified independently. The sheets are assumed either to be welded together or to have free edges separating them. Two methods of solution are employed – the Wiener–Hopf technique and residue calculus, which allows the theoretical development to be verified at various points along the way. The model generalizes the considerable body of published work concerned with wave propagation into and out of floating ice sheets, and across features contained therein such as cracks, open or refrozen leads and embedded icebergs. It can also be applied to breakwaters, very large floating structures and vessels in a seaway. After validation, results are presented showing (a) the details of how the reflection coefficient depends on the geometry of the configuration being modelled; and (b) how a wave energy spectrum evolves as it propagates in a marginal ice zone composed of a large number of identical (coherent) or randomly specified (incoherent) sea-ice plates, as commonly observed in the polar or subpolar oceans.

The stability of two-dimensional thermal convection in an infinite-Prandtl-number fluid layer with zero-stress boundaries is investigated using numerical calculations in three-dimensional rectangles. At low Rayleigh numbers (Ra < 20000) calculations of the stability of two-dimensional rolls to cross-roll disturbances are in agreement with the predictions of Bolton & Busse for a fluid with a large but finite Prandtl number. Within the range 2 × 104 < Ra [les ] 5 × 105, steady rolls with basic wavenumber α > 2.22 (aspect ratio < 1.41) are stable solutions. Two-dimensional rolls with basic wavenumber α < 1.96 (aspect ratio > 1.6) are time dependent for Ra > 4 × 104. For every case in which the initial condition was a time-dependent large-aspect-ratio roll, two-dimensional convection was found to be unstable to three-dimensional convection. Time-dependent rolls are replaced by either bimodal or knot convection in cases where the horizontal dimensions of the rectangular box are less than twice the depth. The bimodal planforms are steady states for Ra [les ] 105, but one case at Ra = 5 × 105 exhibits time dependence in the form of pulsating knots. Calculations at Ra = 105 in larger domains resulted in fully three-dimensional cellular planforms. A steady-state square planform was obtained in a 2.4 × 2.4 × 1 rectangular box. started from random initial conditions. Calculations in a 3 × 3 × 1 box produced steady hexagonal cells when started from random initial conditions, and a rectangular planform when started from a two-dimensional roll. An hexagonal planform started in a 3.5 × 3.5 × 1 box at Ra = 105 exhibited oscillatory time dependence, including boundary-layer instabilities and pulsating plumes. Thus, the stable planform in three-dimensional convection is sensitive to the size of the rectangular domain and the initial conditions. The sensitivity of heat transfer to planform variations is less than 10%.

Elementary calculations indicate that the effect of the Earth's rotation is likely to be important in the dynamics of most internal waves in oceans, lakes and the atmosphere. Here we present measurements of the structure and properties of one class of such waves, namely solitary internal Kelvin waves, in which the Coriolis force generated by wave motion in a stratified fluid is opposed by a pressure gradient and hence change in wave amplitude along its crest. We confirm that the wave speed is independent of the rate at which the system rotates and depends only on the stratification and maximum wave amplitude. However, rotation is shown to have a large effect on both the rate at which the amplitude varies with time and the cross-stream’ structure of the wave. In accordance with well-established theory, the amplitude transverse to the direction of propagation varies exponentially. This results in a decreasing wave speed with increasing distance from the wall, which in turn requires the wave front be curved backwards in order for the wave as a whole to propagate at a speed given by its maximum amplitude. Such a front curvature is not contained within the available theories. The rapid decay of wave amplitude is found to be due to the generation of inertial waves in the homogeneous fluid above and below the internal wave, and a reasonably successful scaling of this effect has been found. We also discuss the adjustment of the waves to geostropic balance and comment on applications of our results to natural systems.

Wave-mean interactions of the classical type, in which the effect of the waves on the mean motion depends on wave breaking or other types of wave dissipation, are to be sharply distinguished from other types of wave-mean interaction that have no such dependence on dissipation. Important cases arise both for unstratified (homentropic) flow and for stably stratified flow under gravity. A very general way of characterizing what is meant by the classical, dissipative type of wave-induced mean motion is to say that the wave-induced mean motions are balanced motions, in a sense to be discussed, and that the effective mean force corresponds to the wave-induced vorticity or potential vorticity transport that results from wave dissipation. For a stratified fluid, ‘potential vorticity’ is to be understood in the sense of Rossby and Ertel. ‘Balanced’ is to be understood in whatever sense is needed to imply the invertibility of the vorticity or potential vorticity field to give the other fields describing the mean motion. At first sight this appears to require that an appropriate Mach, Froude and/or Rossby number for the mean motion should be much smaller than unity, but the fundamental, and in practice less stringent, principal requirement appears to be that the spontaneous emission, or aerodynamic generation, of sound, gravity and/or inertio-gravity waves by the mean flow should be weak.

Three basic examples of dissipative wave-induced mean flow generation are presented and discussed. The first is the transport of vorticity by dissipating sound waves, which gives rise to classical acoustic streaming of the quartz-wind type. The transport or flux of vorticity can always be taken to be an exactly antisymmetric tensor; and in the case of a plane sound wave this tensor fluctuates about a mean value equal to $-\epsilon_{ijk}\dot{q}_k$, where $\dot{q}_k$ is the kth component of $\dot{\boldmath q}$, the rate of dissipation of the pseudomomentum or quasimomentum ${\boldmath q} \approx Ek/\overline{\rho}\omega$ per unit mass. Here $\overline{\rho}$ and E are the mean mass and wave-energy densities, ω the intrinsic frequency, and k the wavenumber. This is a succinct way of making evident why it is only the contribution q to the radiation stress convergence per unit mass that is significant for the generation of mean streaming. The second example is the transport of Rossby–Ertel potential vorticity (PV) by internal gravity waves that are either dissipating laminarly, or ‘breaking’ to produce inhomogeneous three-dimensional turbulence. This PV transport gives rise to mean streaming in much the same way as the vorticity transport in the acoustic example. The transport or flux of PV can always be taken to be directed exactly along the isentropic surfaces θ = constant of the stable stratification, where θ is potential temperature or potential density as appropriate; and in the case of a plane internal gravity wave the wave-induced PV transport fluctuates about a mean value G × q, where G is the basic gradient of θ associated with the stable stratification. This is a succinct way of making evident why it is only the projection of q onto the basic stratification surfaces that is significant. In both the acoustic and the internal-gravity examples the transport is non-advective, and often upgradient. The third example is the corresponding problem for Rossby waves, in which the typical effect of wave dissipation is a downgradient PV transport. This is brought about in an entirely different way, namely through advection of PV anomalies by the fluctuating velocity field of the wave motion, whether the dissipation be laminar or by breaking.

Processes of the sort idealized in the second and third examples are ubiquitous in the Earth's atmosphere and, for instance, largely control the strength of the global-scale middle atmospheric circulation and hence, for instance, the e-folding residence times (∼ 102 y) of man-made chlorofluorocarbons in the lower atmosphere.

We consider a horizontal three-dimensional saturated porous layer, confined in an upright cubic box, heated from below and cooled from above. In the absence of a controller, the fluid maintains a no-motion state for subcritical Rayleigh numbers R < Rc, where Rc depends on the box's aspect ratio. Once this critical number is exceeded, fluid motion ensues. We demonstrate that, with the use of feedback control strategies which suppress flow instabilities, one can maintain a stable no-motion state for Rayleigh numbers far exceeding the classical critical one for the onset of convection. To preserve the equilibrium no-motion state of the classical problem, the controller alters the system's dynamics so as to stabilize an otherwise non-stable state.