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Extreme events in turbulent flow are associated with intense stretching of concentrated vortices, intermittent in both space and time. The occurrence of such events has been investigated in a turbulent flow driven by counter-rotating propellors (Debue et al., J. Fluid Mech., 2021), and local flow structures have been identified. Interesting theoretical problems arise in relation to this work; these are briefly considered in this focus paper.

The behaviour of a viscous drop squeezed between two horizontal planes is treated by both theory and experiment. When the squeezing force F is constant and surface tension is neglected, the theory predicts ultimate growth of the radius a~ t^{1/8}, in excellent agreement with our experiment. Surface tension at the drop boundary is included in the analysis, although negligibly small in the squeezing experiments. The circular evolution is found to be stable under small perturbations. If the force is reversed (F < 0), so that the plates are pulled or levered apart, the boundary of the drop is subject to a fingering instability. The effect of a trapped air bubble at the centre of the drop is then considered. The annular evolution of the drop under constant squeezing is still found to follow a `one-eighth' power law, but this is unstable, the instability originating at the boundary of the air bubble. If the plates are drawn apart, the evolution is still subject to the fingering instability driven from the outer boundary of the annulus. This instability is realised experimentally by levering the plates apart at one corner: fingering develops at the outer boundary and spreads rapidly to the interior as the levering is slowly increased. At a later stage, small cavitation bubbles appear in the very low pressure region far from the point of leverage. This exotic behaviour is discussed in the light of the foregoing theoretical analysis., 23 pages, 12 figures, submitted to J. Fluid Mech

Equations describing the rolling of a spherical ball on a horizontal surface are obtained, the motion being activated by an internal rotor driven by a battery mechanism. The rotor is modeled as a point mass mounted inside a spherical shell and caused to move in a prescribed circular orbit relative to the shell. The system is described in terms of four independent dimensionless parameters. The equations governing the angular momentum of the ball relative to the point of contact with the plane constitute a six-dimensional, nonholonomic, nonautonomous dynamical system with cubic nonlinearity. This system is decoupled from a subsidiary system that describes the trajectories of the center of the ball. Numerical integration of these equations for prescribed values of the parameters and initial conditions reveals a tendency toward chaotic behavior as the radius of the circular orbit of the point mass increases (other parameters being held constant). It is further shown that there is a range of values of the initial angular velocity of the shell for which chaotic trajectories are realized while contact between the shell and the plane is maintained. The predicted behavior has been observed in our experiments.

Based on experimental evidence that vortex reconnection commences with the approach of nearly antiparallel segments of vorticity, a linearised model is developed in which two Burgers-type vortices are driven together and stretched by an ambient irrotational strain field induced by more remote vorticity. When these Burgers vortices are exactly antiparallel, they are annihilated on the strain time-scale, independent of kinematic viscosity $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\nu $ in the limit $\nu \rightarrow 0$. When the vortices are skew to each other, they are annihilated under this action over a local extent that increases exponentially in the stretching direction, with clear evidence of reconnection on the same strain time-scale. The initial helicity associated with the skewed geometry is eliminated during the process of reconnection. The model applies equally to the reconnection of weak magnetic flux tubes under the action of a strain field, when Lorentz forces are negligible.

If, in a large expanse of fluid such as air or water, an object that is heavier than the fluid displaced is released from rest, it descends in a manner that can depend in a complex way on its geometry and density (relative to that of the fluid), and on the fluid viscosity, which, as in other fluid contexts, remains important no matter how small this viscosity may be. A major numerical attack on this problem for the case in which the object is a thin circular disc is presented by Auguste, Magnaudet & Fabre (J. Fluid Mech., vol. 719, 2013, pp. 388–405).

The flow field over an accelerating rotating wing model at Reynolds numbers Re ranging from 250 to 2000 is investigated using particle image velocimetry, and compared with the flow obtained by three-dimensional time-dependent Navier-Stokes simulations. It is shown that the coherent leading-edge vortex that characterises the flow field at Re~200-300 transforms to a laminar separation bubble as Re is increased. It is further shown that the ratio of the instantaneous circulation of the leading-edge vortex in the accel-eration phase to that over a wing rotating steadily at the same Re decreases monotonically with increasing Re. We conclude that the traditional approach based on steady wing rotation is inadequate for the prediction of the aerodynamic performance of flapping wings at Re above about 1000.

The Lighthill–Weis-Fogh ‘clap–fling–sweep’ mechanism for lift generation in insect flight is re-examined. The novelty of this mechanism lies in the change of topology (the ‘break’) that occurs at a critical instanttcwhen two wings separate at their ‘hinge’ point as ‘fling’ gives way to ‘sweep’, and the appearance of equal and opposite circulations around the wings at this critical instant. Our primary aim is to elucidate the behaviour near the hinge point as timetpasses throughtc. First, Lighthill's inviscid potential flow theory is reconsidered. It is argued that provided the linear and angular accelerations of the wings are continuous, the velocity field varies continuously through the break, although the pressure field jumps instantaneously att=tc. Then, effects of viscosity are considered. Near the hinge, the local Reynolds number is very small and local similarity solutions imply a logarithmic (integrable) singularity of the pressure jump across the hinge just before separation, in contrast to the ‘negligible pressure jump’ of inviscid theory invoked by Lighthill. We also present numerical simulations of the flow using a volume penalization technique to represent the motion of the wings. For Reynolds number equal to unity (based on wing chord), the results are in good agreement with the analytical solution. At a realistic Reynolds number of about 20, the flow near the hinge is influenced by leading-edge vortices, but local effects still persist. The lift coefficient is found to be much greater than that in the corresponding inviscid flow.

Shear flow is prone to transient instability in which perturbations having little or no variation in the streamwise direction can grow linearly for a long time if the Reynolds number is large. This behaviour is known to provide a trigger for the development of secondary instabilities and transition to turbulence. It is shown by a simple analysis of Kelvin modes that a spanwise magnetic field is efficient in suppressing this transient instability and therefore in inhibiting the transition to turbulence.

Following part I of this series, the general spinning motion of an axisymmetric rigid body on a horizontal table is further analysed, allowing for slip and friction at the point of contact. Attention is focused on the case of spheroids whose density distribution is such that the centre-of-mass and centre-of-volume coincide. The governing dynamical system is treated by a multiple-scale technique in order to resolve the two time-scales intrinsic to the dynamics. An approximate solution for the high-frequency component of the motion reveals that the normal reaction can oscillate with growing amplitude, and in some circumstances will fall to zero, leading to temporary loss of contact between the spheroid and the table. The exact solution for the free motion that ensues after this ‘jumping’ is analysed, and the time-dependence of the gap between the spheroid and the table is obtained up to the time when contact with the table is re-established. The analytical results agree well with numerical simulations of the exact equations, both up to and after loss of contact.

The viscous interaction of two weakly curved and oppositely directed vortex tubes of flattened cross-section driven together by an imposed uniform strain is considered using a perturbation technique. The evolution of the cross-sectional vorticity distribution is computed, and the rate of reconnexion, as indicated by the decrease of circulation of each tube at the plane of symmetry, is obtained for various values of the Reynolds number and of the (small) aspect ratio of the tubes. Attention is focused particularly on the nonlinear effect of convection by the velocity field induced by the vortices. A similar technique is applied to the problem of reconnexion of magnetic flux tubes, for which the nonlinear effect is different, being that associated with the Lorentz force distribution.

The general spinning motion of an axisymmetric rigid body on a horizontal table is analysed, allowing for slip and friction at the point of contact P. Attention is focused on the case of spheroids (prolate or oblate), and particularly on spheroids whose density distribution is such that the centre-of-mass and centre-of-volume coincide. Four classes of fixed points (i.e. steady states) are identified, and the linear stability properties in each case are determined, assuming viscous friction at P. The governing dynamical system is six-dimensional. Trajectories of the system are computed, and are shown in projection in a three-dimensional subspace; these start near unstable fixed points and (in the case of viscous friction) end at stable fixed points. It is shown inter alia that a uniform prolate spheroid set in sufficiently rapid spinning motion with its axis horizontal is unstable, and its axis rises to a stable steady state, at either an intermediate angle or the vertical, depending on the initial angular velocity. These computations allow an assessment of the circumstances under which the condition described as ‘gyroscopic balance’ is realized. Under this condition, the evolution from an unstable to a stable state is greatly simplified, being described by a first-order differential equation. Oscillatory modes which are stable on linear analysis may be destabilized during this evolution, with consequential oscillations in the normal reaction R at the point of support. The computations presented here are restricted to circumstances in which R remains positive.

A thin-film approximation is used in an analysis of the flow of a thin trickle of viscous fluid down a near-vertical plane. An approximate similarity solution is obtained, representing essentially a source (or sink) flow. Several interpretations of the solution are discussed. Problems concerning the ‘draining’ of viscous films down inclined surfaces have received much attention in the literature. In this paper we use a thin-film approximation to study the steady spreading or contraction of viscous liquid supplied (at a prescribed rate) on a near-vertical plane; gravity may be considered the ‘main’ driving force, but surface tension cannot be neglected. The assumption that the flow changes only slowly down the plane then leads to an approximate similarity solution for this three-dimensional viscous free-surface flow. Several interpretations of this solution are discussed. Only steady flows are considered, so that, in particular, any contact lines are fixed; diculties associated with moving contact lines are thereby avoided. (Cf. Davis, 1983, and the many references therein.) Unsteady flows have been considered (within a thin-film theory) by, for example, Huppert (1982), Schwartz (1989), Lister (1992) and Moriarty et al. (1991).

It is known that two‐dimensional vortices are subject to generic three‐dimensional instabilities. This phenomenon is located near the core of vortices and depends on the eccentricity of their streamlines. In this paper we are concerned with the modification of this instability when stretching is applied to such vortices. We describe this instability by linearizing the Navier–Stokes equations around a basic state, which is an exact time‐dependent solution. The complete system for the perturbations is reduced to a single equation for the perturbed velocity along the vortex span. In the limit of weak stretching, a perturbation theory can be performed and leads to a WKBJ approximation for the solution. This procedure demonstrates that a small amount of stretching is able to prevent the appearance of three‐dimensional instabilities for vortices with a low enough eccentricity. Since most vortices are slightly elliptical in turbulent flows, the above computations are expected to cover a wide range of experimental cases. In particular, it is tentatively argued that this mechanism may explain recent experimental observations [Phys. Fluids 7, 630 (1995)].

International audience; The importance of three-dimensional effects for flapping wings is addressed by means of numerical simulation. In particular, the clap-fling-sweep mechanism is examined. The flow at the beginning of the downstroke is shown to be in reasonable agreement with the two-dimensional approximation. After the wings move farther than one chord length apart, three-dimensional effects become essential. Two values of the Reynolds number are considered. At Re=128, the spanwise flow from the wing roots to the wing tips is driven by the centrifugal forces acting on the mass of the fluid trapped in the recirculation bubble behind the wings. It removes the excess of vorticity and delays the periodic vortex shedding. At Re=1400, vortex breakdown occurs past the outer portion of the wings, and multiple vortex filaments are shed into the wake.

International audience; Numerical simulations of the Lighthill-Weis-Fogh mechanism are performed using a Fourier pseudo-spectral method with volume penalization. Single-winged and double-winged configurations are compared, and the vortex shedding patterns are related to the lift generated in both cases. The computations of the lift coefficient are validated against the results reported previously by Miller and Peskin (J Exp Biol 208:195-212, 2005).

Dynamo action is the process by which kinetic energy is systematically converted to magnetic energy in electrically conducting fluids. This process may occur on a resistive time-scale (i.e. with a growth rate that tends to zero as the magnetic resistivity of the medium tends to zero) in which case the dynamo is described as “slow”; or it may occur on the convective time-scale, in which case the dynamo is described as “fast”. The conventional description of the solar dynamo (an αω dynamo in the terminology of mean-field electrodynamics) places it in the “fast” category, because the usual expressions for the regeneration parameter a and the turbulent resistivity β are both independent of molecular diffusivity η under the conditions obtained deep down in the solar convection zone. Current attempts to construct fast dynamos, and to understand their detailed structure, are therefore of critical importance in the context of solar magnetism.

The Burgers vortex (Burgers 1948) is described by an exact solution of the Navier-Stokes equations in which the effects of uniform axisymmetric stretching are in equilibrium with viscous diffusion. Burgers introduced this vortex as ‘a mathematical model illustrating the theory of turbulence’, and he noted particularly that the vortex had the property that the rate of viscous dissipation per unit length of vortex was independent of viscosity in the limit of vanishing viscosity (i.e. high Reynolds number). This is of course an attractive feature in the light of Kolmogorov’s theory of turbulence, in which the rate of dissipation of energy per unit volume ∈ and the kinematic viscosity v are regarded as independent variables.

The motion of an axisymmetric vortex ring of small cross-section in a viscous incompressible fluid is investigated using the method of matched asymptotic expansions. A general formula for the ring speed is obtained up to third order in є = δ/R 0 (= (v/Г)1/2), the ratio of core to curvature radii, which takes account of the influence of the self-induced strain. Here Г is the circulation and v is the kinematic viscosity of fluid. It is pointed out that the dipole distributed along the centerline of the ring plays a vital role in its movement. Its strength needs be specified at the initial instant in order to remove the indeterminacy of the theory. A new asymptotic development of the Biot-Savart law enables us to calculate the non-local induction velocity at O(є 3)from the dipole. In a special case, we recover Dyson’s inviscid formula (1893). It is demonstrated that the viscosity acts, at O(є 3), to expand the radius of the loop consisting of the stagnation points in the core, when viewed from a certain comoving frame.

The process of relaxation of a unidirectional magnetic field in a highly conducting tenuous fluid medium is considered. Null points of the field play a critical role in this process. During an initial stage of relaxation, variations in magnetic pressure are eliminated, and current sheets build up in the immediate neighborhood of null points. This initial phase is followed by a long diffusive phase of slow algebraic decay of the field, during which fluid is continuously sucked into the current sheets, leading to exponential growth of fluid density and concentration of mass around the null points, which show a tendency to cluster. Ultimately, this second phase of algebraic decay gives way to a final period of exponential decay of the field. The peaks of density at the null points survive as a fossil relic of the decay process. Numerical solution of the governing equations provides convincing confirmation of this three-stage scenario. Generalizations to two- and three-dimensional fields are briefly considered.

Spiral structures are natural candidates for the role of the ‘generic structures’ of turbulent flow, because they are the eventual outcome of Kelvin-Helmholtz instability, an all-pervasive phenomenon associated with nearly all shear flows at very high Reynolds number. Such structures were proposed by Lundgren (1982) in a model of the fine structure of turbulence in which axial stretching of rolled-up spiral vortices played an essential role. This model could be viewed as a natural development of Townsend’s (1951) model of the dissipative structures of turbulence in terms of a random distribution of vortex sheets and/or tubes, each such structure being subjected to the local rate of strain (assumed uniform and constant) associated with all other vortex structures (see Batchelor 1982, § is known that compressive strain (with two positive principle rates of strain) is more likely in isotropic turbulence than extensive strain, so that sheets form with higher probability than tubes. However these are immediately subject to the Kelvin-Helmholtz instability which may be impeded, but not entirely suppressed, by the stretching process.

The low Reynolds number flow between two concentric steadily rotating spheres is considered. The pattern of streamlines is explained in terms of an adiabatic invariant. It is shown that, when the two rotation vectors are not parallel, the streamlines become chaotic when the Reynolds number Re is increased and the onset of global chaos occurs near Re = 20.

The magnetohydrodynamic evolution of axisymmetric magnetic eddies within which the magnetic field is purely toroidal with $B_\theta/r$ piecewise-constant, and the velocity field is poloidal, is studied both analytically and numerically. A family of exact solutions, generalizing Hill's spherical vortex to the case of non-zero magnetic field, is found. These exact solutions are (like Hill's vortex) unstable, so that, under weak disturbance, a narrow spike of vorticity is shed from the neighbourhood of the rear stagnation point. Numerical simulation using a contour-dynamics formulation shows that, for general initial contour shape, a contour singularity appears at a finite time $t^*$ , like that which appears on a disturbed vortex sheet. Techniques of regularization and sample-point redistribution are used so that the eddy contours can be tracked well beyond $t^*$ . When the fluid is initially at rest, the magnetic eddy first contracts towards the axis of symmetry under the action of its Lorentz force distribution; then two spherical fronts form, which propagate in the two opposite directions along the axis of symmetry, in a manner captured well by the exact solution. The magnetic energy remains bounded away from zero despite the fact that there is no topological barrier to its further decrease. Magnetic eddy evolution and the possible existence of steady states under a uniform compressive strain field is also numerically investigated.

We present two classes of exact solutions of the Navier–Stokes equations, which describe steady vortex structures with two-dimensional symmetry in an infinite fluid. The first is a class of similarity solutions obtained by conformal mapping of the Burgers vortex sheet to produce wavy sheets, stars, flowers and other vorticity patterns. The second is a class of non-similarity solutions obtained by continuation and mapping of the classical solution to steady advection–diffusion around a finite circular absorber in a two-dimensional potential flow, resulting in more complicated vortex structures that we describe as avenues, fishbones, wheels, eyes and butterflies. These solutions exhibit a transition from ‘clouds’ to ‘wakes’ of vorticity in the transverse flow with increasing Reynolds number. Our solutions provide useful test cases for numerical simulations, and some may be observable in experiments, although we expect instabilities at high Reynolds number. For example, vortex avenues may be related to counter-rotating vortex pairs in transverse jets, and they may provide a practical means to extend jets from dilution holes, fuel injectors, and smokestacks into crossflows.

Preface 1. Interfacial fluid dynamics Stephen Davis 2. Viscous fingering as an archetype for growth patterns Yves Couder 3. Blood flow in arteries and veins Tim Pedley 4. Open shear flow instabilities Patrick Huerre 5. Turbulence Javier Jimenez 6. Convection in the environment Paul Linden 7. Reflections on magnetohydrodynamics Keith Moffatt 8. Solidification of fluids Grae Worster 9. Geological fluid mechanics Herbert Huppert 10. The dynamic ocean Chris Garrett 11. On global-scale atmospheric and oceanic circulations Michael McIntyre Index.

Kinematic, axisymmetric mean field dynamos are examined for a number of models in order to study the cooperation between small scale turbulent motions (α-affect) and the large scale motions (ω-effect and meridional flow). For a spherical dynamo we show that it is more difficult to excite an α2-dynamo with dipolar parity in the presence of differential rotation, which increases with increasing depth, when the magnitude of the toroidal shearing effect is too small. Meridional streaming in the absence of differential rotation has an inhibitory effect on α2-dynamos. For dynamos in spheres the ease with which a dynamo is excited when the combined effects of differential rotation and meridional flow are present depends on the model for ω(r,θ). For a model where the maximum in the toroidal shearing is towards the edge of the region enclosing the streamlines a small amount of meridional flow enhances dynamo action. For dynamos confined to a spherical shell the effect of a small amount of single cell meri...

SUMMARY The axial diffusion of a passive scalar field (e.g. temperature) in Poiseuille flow through a duct is considered, taking account of leakage of heat through the duct boundary. The cases of the two-dimensional channel and the pipe of circular cross-section are considered in detail, and it is shown that (i) the centroid of the scalar field moves (asymptotically) with a velocity intermediate between the mean and the maximum flow rates and increases with increasing wall conductance, and (ii) the effective diffusivity in the flow direction is a decreasing function of wall conductance. The temperature field downstream of a maintained heat source is determined as a function of wall conductance.

Steady solutions of the Euler equations for flow of an inviscid incompressible fluid may be obtained by considering the process of magnetic relaxation to analogous magnetostatic equilibria in a viscous perfectly conducting fluid. In particular, solutions which represent rotational disturbances propagating without change of structure in an unbounded fluid may be obtained by this method. When conditions are axisymmetric, these disturbances are vortex rings of general structure, which may include a swirl component of velocity. This situation is analysed in some detail, and it is shown that the vortex is characterised by two functions: V(ψ), the volume within toroidal surfaces ψ = cst. and W(ψ), the toroidal volume flux inside the torus ψ = cst. For each choice of {V(ψ), W(ψ)}, satisfying appropriate limit conditions, there exists at least one vortex ring of steady structure.

This paper is a summary of the Second Bat-Sheva Seminar on magnetohydrodynamic (MHD) Flows and turbulence. It was held in the University of the Negev, Israel, on 28-31 March 1978, with 64 participants from 7 countries. Reviews and research papers were presented on the general theory of MHD, MHD duct flows (with emphasis on novel aspects such as non-uniform fields and fluid properties, bends, free-surface effects and longitudinal diffusion), two-phase flows (especially those likely to occur in a liquid-metal generator), turbulence and instabilities, and electrically driven flows (with new results presented for the theory of laminar and turbulent flows in induction furnaces, and for the theory of thermo-electrically driven flows in transverse magnetic fields). One day of the conference was devoted to turbulence, mainly without magnetic fields, with reviews and new results presented on the theory and measurements of coherent structures, intermittency at high Reynolds number, methods of calculating shear flows, and measurement techniques. The seminar was a strange mixture of people and topics, which produced some interesting papers and some useful discussion.

The effect of an applied transverse magnetic field on the development of a two-dimensional jet of incompressible fluid is examined. The jet is prescribed in terms of its mass flux ρQ0 and its lateral scale d at an initial section x = 0. The three dimensionless numbers characterizing the problem are a Reynolds number R = Q0/ν, a magnetic Reynolds number Rm = μσQ0, and a magnetic interaction parameter N = σB20d2/ρQ0, where ρ represents density, σ conductivity, μ permeability and B0 applied field strength, and it is assumed that \[ R_m \ll 1,\quad R\gg 1,\quad N\ll 1. \] It is shown that when M2 = RN [Gt ] 1, an inviscid treatment is appropriate, and that the effect of the magnetic field is then to destroy the jet momentum within a distance of order N−1 in the downstream direction. A general solution for inviscid development is obtained, and it is shown that a large class of velocity profiles (though not all of them) are self-preserving.When M2 [Lt ] 1, it is shown that the viscous similarity solution obtained by Moreau (1963a, b) is relevant. This solution is re-derived and re-interpreted; it implies that the jet momentum is destroyed within a distance of order $R^{\frac{1}{4}}N^{-\frac{3}{4}}$ in the downstream direction.Some further aspects of the jet annihilation problem are qualitatively discussed in § 4, viz. the nature of the overall flow field, the effect of the presence of distant boundaries, the effect of increasing Rm to order unity and greater, and the effect of oblique injection. Finally the development of a jet of conducting fluid into a nonconducting environment is considered; in this case the jet is not stopped by the magnetic field unless a return path outside the fluid for the induced current is available.

The interior of an insulating cylindrical container is supposed filled with an incompressible, electrically conducting, viscous fluid. An externally applied magnetic field is caused to rotate uniformly about an axis parallel to the cylinder generators (by applying two alternating components out of phase at right angles). Induced currents in the fluid give rise to a Lorentz force which drives a velocity field, which in general may have a steady and a fluctuating component. The particular case of a circular cylindrical container in a transverse magnetic field is studied in detail. Under certain reasonable assumptions, the resulting flow is shown to have only the steady component, and the distribution of this component is determined. Some conjectures are offered about the stability of this flow and about the corresponding flows in cavities of general shape.

It is shown that in an incompressible fluid in which the magnetic diffusivity λ is much less than the kinematic viscosity ν, certain magnetic field distributions of limited spatial extent (conveniently described as magnetic eddies) can exist on a length scale such that the associated Reynolds number and magnetic Reynolds number are respectively small and large compared with unity. The Lorentz forces are in equilibrium with the dynamic forces associated with the fluid motion. The boundary condition imposed on this motion is that at a large distance from a magnetic eddy the velocity field should be a uniform axisymmetric irrotational straining motion. The eddies are steady in the limit λ → 0, but decay slowly in a fluid of finite conductivity. Two particular eddies are considered in detail: a disk-shaped eddy in a compressive straining motion, and a spherical eddy in an extensive straining motion. Possible applications to turbulence in interstellar gas clouds are qualitatively considered.

If the magnetic diffusivity of a fluid in turbulent motion is greater than its kinematic viscosity, then, according to Batchelor, any magnetic field initially present must ultimately decay to zero. However, if a magnetic field is externally maintained, the turbulence will generate fluctuations that may be quite large if the magnetic Reynolds number is large compared with unity. The spectrum of these fluctuations increases up to a wave number kc marking the threshold of conduction effects, and falls off rapidly beyond kc. The net effect of the turbulence can be expressed in terms of an eddy conductivity equal to the molecular conductivity multiplied by the (−5/2)th power of the magnetic Reynolds number.

The magnetic field of the Earth is generated by dynamo action associated with the upwelling of buoyant material in the liquid outer core. It is argued that this upwelling occurs in the form of mushroom-shaped blobs of material released from the mushy zone at the inner core boundary (ICB), and having a very small density defect δρ/ρ. The rise of buoyant material with velocity w is compensated by the slow rate of growth of the solid inner core. The resulting mass balance, combined with approximate geostrophic force balance in the core leads to estimates $$\delta \rho /\rho \sim 3 \times {10^{ - 9}},w \sim 2 \times {10^{ - 4}}m/s.$$ Each rising blob drives a Taylor column, and the helicity and α-effect associated with this flow is estimated. A mean-field dynamo driven by this α-effect in conjunction with differential rotation generates a magnetic field whose strength is determined in order of magnitude by the plausible assumption of magnetostrophic equilibrium.

The flow of electric currents between electrodes immersed in a conducting fluid permeated by a uniform magnetic field is considered. The current interacts with the field to generate a velocity field which modifies the steady current flow pattern. The magnetic Reynolds number of the motion is assumed small, and inertia forces are assumed negligible compared with either magnetic or viscous forces. Under these circumstances the flow pattern depends only on the Hartmann number M in addition to the boundary conditions imposed. When M → ∞, layers of discontinuity may develop in the fluid. Across these layers the tangential component of electric field is discontinuous, the normal component within the layer being of order δ−1 where δ is the layer thickness. Two particular geometries are discussed in detail. In the first, the magnetic field is perpendicular to the electrodes, and a layer of discontinuity, with the character of a non-spreading jet, develops as M → ∞. In the second, the magnetic field is parallel to the electrodes, and the discontinuities that develop in the limit are confined to the boundaries of the fluid.

The response of a Hill’s spherical vortex to an irrotational axisymmetric small perturbation is examined on the assumption that viscous diffusion of vorticity is negligible. The problem of determining the response is first reduced to a system of differential equations for the evolution of the Legendre coefficients of the disturbance stream function. This system is then solved approximately and it is shown that, if the initial disturbance is such as to make the vortex prolate spheroidal, the vortex detrains a fraction $c of its original volume, where e is the fractional extension of the axis of symmetry in the imposed distortion. The detrained fluid forms a thin spike growing from the rear stagnation point. If the vortex is initially oblate, irrotational fluid is entrained a t the rear stagnation point to the interior of the vortex.

The deflection of a thin, two-dimensional stream of liquid metal due to external high-frequency currents is investigated. A simple theoretical model assuming uni-directional flow is presented. The relationship between the angle of deflection of the stream and the power supplied to the perturbing currents is determined. Experiments are performed in which a free-falling mercury sheet is deflected by two anti-parallel line currents. The agreement between theory and experiment is reasonable, despite a tendency towards three-dimensionality in the latter.

It is shown that a random superposition of inertial waves in a rotating conducting fluid can act as a dynamo, i.e. can systematically transfer energy to a magnetic field which has no source other than electric currents within the fluid. Dynamo action occurs provided the statistical properties of the velocity field lack reflexional symmetry, and this occurs when conditions are such that there is a net energy flux (positive or negative) in the direction of the rotation vector Ω.If the magnetic field grows from an infinitesimal level, then the mode of maximum growth rate dominates before the back-reaction associated with the Lorentz force becomes significant. This mode is first determined, and then the back-reaction associated with it alone is analysed. It is shown that the magnetic energy grows exponentially during the stage when the Lorentz forces are negligible, then reaches a maximum depending on the values of the parameters \[ R_m = u_0 l/\lambda,\quad Q = \Omega l^2/\lambda, \] (u0 = initial r.m.s. velocity, l = length scale characteristic of the velocity field, λ = magnetic diffusivity) and ultimately decays as t−1 (equation (5.15)). This decay is coupled with a decay of the velocity field due to ohmic dissipation, and it occurs because there is no external source of energy for the fluid motion.

Low magnetic Reynolds number magnetohydrodynamic (MHD) flows and low Rossby number rotating flows share a number of common features. Both are subjected to a strong linear force: Lorentz or Coriolis. From an energetic point of view, they are very different, as Lorentz forces are dissipative in nature (Joule dissipation adding to viscous dissipation) while Coriolis forces are purely conservative. Both forces however tend to favour a two-dimensional (2D) flow, independent of the direction of the applied magnetic field or rotation axis.