New York, 1995,406 + xi pp., ISBN 3-540 60258-5, DM 86.00 hardback.Most magnetic fields observed in nature are produced by electric currents, as in the case of the main geomagnetic field associated with electric currents in the Earth's liquid metallic core. Of the possible causes of the electromotive forces needed to drive these currents, motional induction involving movement of conducting material in a magnetic field is usually much more potent and persistent than chemical, thermoelectric and other effects. Larmor appears to have been the first to appreciate the quantitative importance of the so-called 'self-exciting dynamo mechanism' when in 1919 he proposed that the magnetic field of the Sun is produced by motional induction involving the movement of fluid in the convecting regions of the solar interior.Motions in electrically conducting fluids can-if they are sufficiently rapid and complicated-stretch, twist and fold magnetic lines of force, just as a baker manipulates dough, and a topologist operates on mathematical curves in multidimensional space. This dynamo process is now invoked in the interpretation of cosmical magnetic fields, such as those of the Earth and other planets, the Sun and other stars, and indeed of the whole Galaxy. Stretching, twisting and folding redistributes magnetic field lines and creates new ones, thereby amplifying magnetic energy at the expense of kinetic energy of fluid motion, which is usually driven by buoyancy forces due to the action of gravity on density inhomogeneities.Dynamo theory is based on the highly non-linear equations of magnetohydrodynamics, which express the laws of mechanics, electrodynamics and thermodynamics applied to a continuous medium. The most extensive studies in the subject are those of 'kinematic dynamos', in which for reasons of mathematical expediency the governing equations are simplified by specifying the velocity field in the equations of electrodynamics, thereby obviating the need to include the other equations. More recent work, however, includes attempts to obtain simultaneous solutions for both the magnetic and velocity fields using the full set of equations. The whole subject suffers from the lack of useful experimental guidance, for with the fluids available it is impossible to reproduce the processes involved on the small scale of the laboratory.The first significant finding of kinematic dynamo theory was the Cowling theorem of 1934, to the effect that no magnetic field that retains an axis of symmetry can be maintained by motional induction against the effects of ohmic dissipation. Immediate progress with the development of Larmor's idea would have followed from Cowling's work had mathematicians of the day responded to the direct challenge of seeking existence theorems for non-axisymmetric magnetic fields. In the event, more than a decade was to elapse before the dynamo idea was taken up seriously (by Elsasser, Frenkel and Bullard), and nearly a quarter of a century before Backus and Herzenberg (independently in 1958) provided the first exi...

The relaxation of a smooth two-dimensional vortex to axisymmetry, also known as 'axisymmetrization', is studied asymptotically and numerically. The vortex is perturbed at t = 0 and differential rotation leads to the wind-up of vorticity fluctuations to form a spiral. It is shown that for infinite Reynolds number and in the linear approximation, the vorticity distribution tends to axisymmetry in a weak or coarse-grained sense: when the vorticity field is integrated against a smooth test function the result decays asymptotically as t −λ with λ = 1 + (n 2 + 8) 1/2 , where n is the azimuthal wavenumber of the perturbation and n > 1. The far-field stream function of the perturbation decays with the same exponent. To obtain these results the paper develops a complete asymptotic picture of the linear evolution of vorticity fluctuations for large times t, which is based on that of Lundgren (1982). IntroductionIn fluid flow at high Reynolds number there is a tendency for vorticity to aggregate to form coherent vortices, both for planar flows (e.g. McWilliams 1984McWilliams , 1990Benzi et al. 1986;Brachet et al. 1988) and in three dimensions (e.g. Kuo & Corrsin 1971, 1972Siggia 1981;Kerr 1985;She, Jackson & Orszag 1990;Vincent & Meneguzzi 1991). Such concentrations of vorticity are associated with differential rotation of the fluid, both inside and outside the vortex. This causes stretching of fluid elements and means that fluctuations of vorticity (or a passive scalar) are wound up into characteristic spiral structures, and so are driven to small scales. Our goal is to discuss some of the consequences of differential rotation and this winding-up process and its implications for the behaviour of vorticity and scalars in coherent planar vortices.We consider an idealized problem in which a smooth axisymmetric vortex with vorticity Ω = Ω 0 (r) and associated stream function Ψ = Ψ 0 (r) is perturbed so as to gain small non-axisymmetric components, and we study their subsequent evolution. If this event occurs at t = 0 then for t > 0 the total vorticity and stream function may be written

Saffman argues that in decaying two-dimensional turbulence approximate dis-continuities of vorticity will form, and the energy spectrum will fall off as k−4 Saffman assumes that these discontinuities are well separated; in this paper, we examine how accumulation points of such discontinuities may give an energy spectrum of between k−4 and k−3. In particular we examine the energy spectra of spiral structures which form round the coherent vortices that are observed in numerical simulations of decaying two-dimensional turbulence. If the filaments of the spiral are assumed to be passively advected, the instantaneous energy spectrum has a $k^{-11/3}$ range. Thus we come some way to reconciling the argument of Saffman and the k−3 energy spectrum predicted by models of quasi-equilibrium two-dimensional turbulence based on a cascade of enstrophy in Fourier space.

Many fluctuation-driven phenomena in fluids can be analysed effectively using the generalised Lagrangian mean (GLM) theory of Andrews & McIntyre (1978a). This finiteamplitude theory relies on particle-following averaging to incorporate the constraints imposed by the material conservation of certain quantities in inviscid regimes. Its original formulation, in terms of Cartesian coordinates, relies implicitly on an assumed Euclidean structure; as a result, it does not have a geometrically intrinsic, coordinate-free interpretation on curved manifolds, and suffers from undesirable features. Motivated by this, we develop a geometric generalisation of GLM that we formulate intrinsically using coordinate-free notation. One benefit is that the theory applies to arbitrary Riemannian manifolds; another is that it establishes a clear distinction between results that stem directly from geometric consistency and those that depend on particular choices.Starting from a decomposition of an ensemble of flow maps into mean and perturbation, we define the Lagrangian-mean momentum as the average of the pull-back of the momentum one-form by the perturbation flow maps. We show that it obeys a simple equation which guarantees the conservation of Kelvin's circulation, irrespective of the specific definition of the mean flow map. The Lagrangian-mean momentum is the integrand in Kelvin's circulation and distinct from the mean velocity (the time derivative of the mean flow map) which advects the contour of integration. A pseudomomentum consistent with that in GLM can then be defined by subtracting the Lagrangian-mean momentum from the one-form obtained from the mean velocity using the manifold's metric.The definition of the mean flow map is based on choices made for reasons of convenience or aesthetics. We discuss four possible definitions: a direct extension of standard GLM, a definition based on optimal transportation, a definition based on a geodesic distance in the group of volume-preserving diffeomorphisms, and the glm definition proposed by Soward & Roberts (2010). Assuming small-amplitude perturbations, we carry out orderby-order calculations to obtain explicit expressions for the mean velocity and Lagrangianmean momentum at leading order. We also show how the wave-action conservation of GLM extends to the geometric setting.To make the paper self-contained, we introduce in some detail the tools of differential geometry and main ideas of geometric fluid dynamics on which we rely. These include variational formulations which we use for alternative derivations of some key results. We mostly focus on the Euler equations for incompressible inviscid fluids but sketch out extensions to the rotating-stratified Boussinesq, compressible Euler, and magnetohydrodynamic equations. We illustrate our results with an application to the interaction of inertia-gravity waves with balanced mean flows in rotating-stratified fluids.

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