H. Keith Moffatt

Magnetostrophic Turbulence and the Geodynamo

The flow generated by a random buoyancy field in a rotating medium permeated by a dynamo-generated magnetic field is considered, under the assumptions that the Rossby number and the magnetic Reynolds number (based on the scale of the buoyancy fluctuations) are both small. This permits linearisation of the governing evolution equations. Provided ‘up-down’ symmetry is broken, a mean helicity and an associated α-effect are generated. These are calculated in terms of the spectrum function of the buoyancy field. Expressions are also obtained for the buoyancy flux and the Reynolds stresses (kinetic and magnetic), and an outline dynamo scenario is proposed. The nature of this type of magnetostrophic turbulence is briefly discussed.

SMALL-SCALE HYDROMAGNETIC FLOW IN THE EARTH'S CORE: RISE OF A VERTICAL BUOYANT PLUME

Abstract The steady and transient flow induced by a vertical cylinder of buoyant electrically conducting fluid immersed in an infinite extent of slightly denser fluid in the presence of a horizontal magnetic field is investigated, with the aim of elucidating the small-scale flow within the Earths core. The evolution from a state of rest may be divided into three regimes. For short times [t O(L 2/v)] lateral viscous diffusion also becomes important and a quasi-steady state is reached...

Soap-film Möbius strip changes topology with a twist singularity

It is well-known that a soap film spanning a looped wire can have the topology of a Möbius strip and that deformations of the wire can induce a transformation to a two-sided film, but the process by which this transformation is achieved has remained unknown. Experimental studies presented here show that this process consists of a collapse of the film toward the boundary that produces a previously unrecognized finite-time twist singularity that changes the linking number of the film’s Plateau border and the centerline of the wire. We conjecture that it is a general feature of this type of transition that the singularity always occurs at the surface boundary. The change in linking number is shown to be a consequence of a viscous reconnection of the Plateau border at the moment of the singularity. High-speed imaging of the collapse dynamics of the film’s throat, similar to that of the central opening of a catenoid, reveals a crossover between two power laws. Far from the singularity, it is suggested that the collapse is controlled by dissipation within the fluid film surrounding the wire, whereas closer to the transition the power law has the classical form arising from a balance between air inertia and surface tension. Analytical and numerical studies of minimal surfaces and ruled surfaces are used to gain insight into the energetics underlying the transition and the twisted geometry in the neighborhood of the singularity. A number of challenging mathematical questions arising from these observations are posed.

Topological constraints and their breakdown in dynamical evolution

A variety of physical and biological systems exhibit dynamical behaviour that has some explicit or implicit topological features. Here, the term ‘topological’ is meant to convey the idea of structures, e.g. physical knots, links or braids, that have some measure of invariance under continuous deformation. Dynamical evolution is then subject to the topological constraints that express this invariance. The simplest problem arising in these systems is the determination of minimum-energy structures (and routes towards these structures) permitted by such constraints, and elucidation of mechanisms by which the constraints may be broken. In more complex nonequilibrium cases there can be recurring singularities associated with topological rearrangements driven by continuous injection of energy. In this brief overview, motivated by an upcoming program on ‘Topological Dynamics in the Physical and Biological Sciences’ at the Isaac Newton Institute for Mathematical Sciences, we present a summary of this class of dynamical systems and discuss examples of important open problems.

Interpretation of Invariants of the Betchov-Da Rios Equations and of the Euler Equations

Interest in the study of invariant quantities is generally motivated by the need to interpret and to understand their meaning and their fundamental role in the theory. The invariants we shall consider in this paper emerge in two contexts. In the context of the localized induction approximation (LIA) for the motion of an inextensible vortex filament in a perfect fluid flow, we shall deal with certain conserved quantities that emerge from the BetchovDa Rios equations; the polynomial invariants for the related nonlinear Schrodinger equation (NLSE) are calculated employing the recurrence formula of Zakharov and Shabat (1974); these quantities are constants of the motion for the vortex as long as self-intersection does not occur and long-distance effects are neglected. Some of them are then interpreted in terms of kinetic energy, momentum, angular momentum, “pseudo-helicity,” and some inequalities between physical and global geometrical properties are stated. In the context of the Euler equations, given that the topology of the vortex structure is conserved and the helicity invariant is the natural measure of the topological complexity of the field (Moffatt, 1969; Arnol’d, 1974), we comment further on this role of helicity; moreover, by associating an “energy spectrum” with any knot or link present in the fluid, the lowest ground-state enstrophy is interpreted as an even more powerful invariant for evaluating the knot or link complexity of the field structure.

Local and Global Perspectives in Fluid Dynamics

The smoothness, or alternatively the finite-time singularity, of the Navier-Stokes equations offers a challenge that will continue to make great demands on both analytical ingenuity and computational power. If, as computer simulations continue to indicate (Kerr 1997), a finite-time singularity does occur and if this is generic behavior, then of course we shall have to understand by what mechanism these putative singularities are resolved. Since the pressure gradient must also become unbounded as a singularity is approached, the incompressibility assumption, on which most analyses of this phenomenon are based, becomes no longer tenable. The infinite stress at a singularity can be relieved by cavitation in liquids, and by acoustic radiation in gases. The Japanese bath provides a congenial environment for the contemplation of such problems!

Helicity—invariant even in a viscous fluid

Observing and probing writhe and twist in vortex dynamics The vortex ring is a fundamental phenomenon of fluid dynamics, recognized since the seminal investigations of Helmholtz (1) and Kelvin (2). Its familiar manifestation as a “smoke ring” in air derives from the fact that both smoke and vorticity (local fluid spin) are transported with the flow, which is “induced” by the vortex itself; so the smoke provides a natural visualization of the vorticity (see the photo). Vortex rings can also be generated in water and visualized either by dye or by small air bubbles that migrate to the low-pressure region at the core of the vortex. On page 487 of this issue, Scheeler et al. (3) explore a particular property of a vortex ring whose core is helical rather than circular in form. This property, helicity, is an integral over the fluid domain that expresses the correlation between velocity and vorticity, and an invariant of the classical Euler equations of ideal (inviscid) fluid flow. The question addressed by Scheeler et al. is the extent to which the helicity remains invariant when fluid viscosity, unavoidable in reality, is taken into consideration.

FOREWORD: Turbulence Colloquium Marseille 2011

Turbulence remains one of the oldest and most challenging research problems in both pure and applied science; recognition of the phenomenon can be traced back to Leonardo da Vinci who introduced the word ‘Turbolenza’. Turbulence characterises the flow of a fluid (gas, liquid or plasma) dominated by nonlinear interactions that make its description and the prediction of its evolution exceptionally difficult. Turbulence has been studied for several centuries by mathematicians as well as by physicists and engineers. It is still an open problem since no satisfactory theory is yet available, from either mathematical or physical viewpoint. Moreover, the turbulent regime may not be as universal as commonly supposed. In 1961, Alexandre Favre organised one of the earliest international colloquia dedicated to turbulence on the occasion of the inauguration of the ‘Institut de Mecanique Statistique de la Turbulence (IMST)’ that he founded in Marseille. The impact of this colloquium was crucial and continues to play an important role 50 years later. It was attended by outstanding luminaries of the subject:Kolmogorov,Yaglom, vonKarman,G.I. Taylor, Liepmann, Laufer, Corrsin, Batchelor, Kovasznay, Kraichnan and many others. Key problems were identified and presented during review lectures given by several invited speakers, followed by extended open discussions [1]. This colloquium led to the development of research areas that are still very much alive to this day.