The particle paths of the Arnold-Beltrami-Childress (ABC) flows \[ u = (A \sin z+ C \cos y, B \sin x + A \cos z, C \sin y + B \cos x). \] are investigated both analytically and numerically. This three-parameter family of spatially periodic flows provides a simple steady-state solution of Euler's equations. Nevertheless, the streamlines have a complicated Lagrangian structure which is studied here with dynamical systems tools. In general, there is a set of closed (on the torus, T3) helical streamlines, each of which is surrounded by a finite region of KAM invariant surfaces. For certain values of the parameters strong resonances occur which disrupt the surfaces. The remaining space is occupied by chaotic particle paths: here stagnation points may occur and, when they do, they are connected by a web of heteroclinic streamlines.When one of the parameters A, B or C vanishes the flow is integrable. In the neighbourhood, perturbation techniques can be used to predict strong resonances. A systematic search for integrable cases is done using Painlevé tests, i.e. studying complex-time singularities of fluid-particle trajectories. When ABC ≠ 0 recursive clustering of complex time singularities occurs that seems characteristic of non-integrable behaviour.

The linear stability of convection in a rapidly rotating sphere studied here builds on well established relationships between local and global theories appropriate to the small Ekman number limit. Soward (1977) showed that a disturbance marginal on local theory necessarily decays with time due to the process of phase mixing (where the spatial gradient of the frequency is non-zero). By implication, the local critical Rayleigh number is smaller than the true global value by an O(1) amount. The complementary view that the local marginal mode cannot be embedded in a consistent spatial WKBJ solution was expressed by Yano (1992). He explained that the criterion for the onset of global instability is found by extending the solution onto the complex s-plane, where s is the distance from the rotation axis, and locating the double turning point at which phase mixing occurs. He implemented the global criterion on a related two-parameter family of models, which includes the spherical convection problem for particular O(1) values of his parameters. Since he used one of them as the basis of a small-parameter expansion, his results are necessarily approximate for our problem.Here the asymptotic theory for the sphere is developed along lines parallel to Yano and hinges on the construction of a dispersion relation. Whereas Yano's relation is algebraic as a consequence of his approximations, ours is given by the solution of a second-order ODE, in which the axial coordinate z is the independent variable. Our main goal is the determination of the leading-order value of the critical Rayleigh number together with its first-order correction for various values of the Prandtl number.Numerical solutions of the relevant PDEs have also been found, for values of the Ekman number down to 10−6; these are in good agreement with the asymptotic theory. The results are also compared with those of Yano, which are surprisingly good in view of their approximate nature.

The correct asymptotic theory for the linear onset of instability of a Boussinesq fluid rotating rapidly in a self-gravitating sphere containing a uniform distribution of heat sources was given recently by Jones et al. (2000). Their analysis confirmed the established picture that instability at small Ekman number $E$ is characterized by quasi-geostrophic thermal Rossby waves, which vary slowly in the axial direction on the scale of the sphere radius $r_o$ and have short azimuthal length scale $O(E^{1/3}r_o)$. They also confirmed the localization of the convection about some cylinder radius $s\,{=}\,s_M$ roughly $r_o/2$. Their novel contribution concerned the implementation of global stability conditions to determine, for the first time, the correct Rayleigh number, frequency and azimuthal wavenumber. Their analysis also predicted the value of the finite tilt angle of the radially elongated convective rolls to the meridional planes. In this paper, we study small-Ekman-number convection in a spherical shell. When the inner sphere radius $r_i$ is small (certainly less than $s_M$), the Jones et al. (2000) asymptotic theory continues to apply, as we illustrate with the thick shell $r_i\,{=}\,0.35\,r_o$. For a large inner core, convection is localized adjacent to, but outside, its tangent cylinder, as proposed by Busse & Cuong (1977). We develop the asymptotic theory for the radial structure in that convective layer on its relatively long length scale $O(E^{2/9}r_o)$. The leading-order asymptotic results and first-order corrections for the case of stress-free boundaries are obtained for a relatively thin shell $r_i\,{=}\,0.65\,r_o$ and compared with numerical results for the solution of the complete PDEs that govern the full problem at Ekman numbers as small as $10^{-7}$. We undertook the corresponding asymptotic analysis and numerical simulation for the case in which there are no internal heat sources, but instead a temperature difference is maintained between the inner and outer boundaries. Since the temperature gradient increases sharply with decreasing radius, the onset of instability always occurs on the tangent cylinder irrespective of the size of the inner core radius. We investigate the case $r_i\,{=}\,0.35\,r_o$. In every case mentioned, we also apply rigid boundary conditions and determine the $O(E^{1/6})$ corrections due to Ekman suction at the outer boundary. All analytic predictions for both stress-free and rigid (no-slip) boundaries compare favourably with our full numerics (always with Prandtl number unity), despite the fact that very small Ekman numbers are needed to reach a true asymptotic regime.

The existence of fast dynamos caused by steady motion of an electrically conducting fluid is established by consideration of a two-dimensional spatially periodic flow: the velocity, which is independent of the vertical coordinate z, is finite and continuous everywhere but the vorticity is infinite at the X-type stagnation points. A mean-field model is developed using boundary-layer methods valid in the limit of large magnetic Reynolds number R. The magnetic field is confined to sheets, width of order R−½. The mean magnetic field lies and is uniform on horizontal planes: its direction is independent of time but rotates once about the vertical axis over a short distance 2πl, where l−1 = R½β and β is a vertical stretched wavenumber independent of R. Its alternating direction gives it a rope-like structure within the sheets. An α-effect is calculated for the model, whose strength for a given flow is a function of β and R. Two sources of α-effect are isolated whose relative importance depends critically on the size of β. When the vorticity is finite everywhere and β [Lt ] 1, the dynamo is ‘almost’ fast with growth rates of order (ln R)−1. The maximum growth rate ln (ln R)/ln R occurs when, correct to leading order, β is (ln R)−½. The asymptotic results valid for large R compare excellently with Roberts (1972) modal analysis for finite R.

Using the first order smoothing approximation and a iiovel technique (double Fourier transformation and expansion) a number of results, new and old, in the theory of mcan field electrodynamics and magnetohyclrodynamics are given a systematic and general derivation. They are cast into forms which bring into new prominence the role of the helicity spectrum in induction processes. The situations in which the results may be cxpccted to be accurate are delineated. New expressions are given for the reduction in the mean electromotive force created by the LORENTZ forces acting on the microscalc turbulence.Mit Hilfe der Korrelationsapproximation zweiter Ordiiung und eincr neuen Technik (doppelte Fouriertransformation und -entwicklung) wird einc Reihe von neuen und alten Ergebnissen in der Theorie der Elektrodynamik gemittelter Felder und der Magnetohydrodynamik durch eiiie systematische und allgemeine Ableitung dargestellt. Sie werden in Formen gebracht, die die Rolle des Helicity-Spektrums in Induktionsprozessen in neuem Licht erscheinen lassen. Die Situationen, in denen die Ergebnisse exakt gIiltig sind, werden angcgeben. Neue Ausdrtkke fQr die Bestimmung der mittleren elektromotorischen Kraft. die durch die Wirkung der LoRENTz-Krtifte auf die Turbulenz hervorgerufen werden, werden mitgeteilt.