Migration of an insulating particle under the action of uniform ambient electric and magnetic fields. Part 1. General theory

International audienceThe behaviour of an insulating particle suspended in a liquid metal and subject to the influence of locally uniform electric and magnetic fields (E,B) is considered. The electric field drives a current J which is perturbed by the presence of the particle, and the resulting Lorentz force drives a flow. It is assumed that both the Reynolds number and the Hartmann number based on particle size are small. If the particle is fixed, it experiences a force and couple that are each bilinear in J and B; if it is freely suspended, then it moves with translational velocity U and angular velocity O each similarly bilinear in J and B. The general form of these bilinear relationships is determined, with particular attention to three types of particle symmetry: (i) isotropy; (ii) axisymmetry; and (iii) orthotropy

Evolution of the Leading-Edge Vortex over an Accelerating Rotating Wing

AbstractThe 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 alpha-effect and current helicity for fast sheared rotators

We explore the alpha-effect and the small-scale current helicity, for the case of weakly compressible magnetically driven turbulence that is subjected to the differential rotation. No restriction is applied to the amplitude of angular velocity, i.e., the derivations presented are valid for an arbitrary Coriolis number, though the differential rotation itself is assumed to be weak. The expressions obtained are used to explore the possible distributions of alpha-effect and current helicity in convection zones (CZ) of the solar-type stars. The implications of the obtained results to the mean-field dynamo models are discussed.Comment: 20 pages, 6 figure

Evidence for topological nonequilibrium in magnetic configurations

We use direct numerical simulations to study the evolution, or relaxation, of magnetic configurations to an equilibrium state. We use the full single-fluid equations of motion for a magnetized, non-resistive, but viscous fluid; and a Lagrangian approach is used to obtain exact solutions for the magnetic field. As a result, the topology of the magnetic field remains unchanged, which makes it possible to study the case of topological nonequilibrium. We find two cases for which such nonequilibrium appears, indicating that these configurations may develop singular current sheets.Comment: 10 pages, 5 figure

Dynamics of the Tippe Top via Routhian Reduction

We consider a tippe top modeled as an eccentric sphere, spinning on a horizontal table and subject to a sliding friction. Ignoring translational effects, we show that the system is reducible using a Routhian reduction technique. The reduced system is a two dimensional system of second order differential equations, that allows an elegant and compact way to retrieve the classification of tippe tops in six groups as proposed in [1] according to the existence and stability type of the steady states.Comment: 16 pages, 7 figures, added reference. Typos corrected and a forgotten term in de linearized system is adde

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations

We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal Transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized Hydrostatic-Boussinesq equations) to various models involving Optimal Transport (and the related Monge-Ampere equation. This includes the 2D semi-geostrophic equations and some fully non-linear versions of the so-called high-field limit of the Vlasov-Poisson system and of the Keller-Segel for Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology

Three-dimensional stability of Burgers vortices

Burgers vortices are explicit stationary solutions of the Navier-Stokes equations which are often used to describe the vortex tubes observed in numerical simulations of three-dimensional turbulence. In this model, the velocity field is a two-dimensional perturbation of a linear straining flow with axial symmetry. The only free parameter is the Reynolds number $Re = \Gamma/\nu$, where $\Gamma$ is the total circulation of the vortex and $\nu$ is the kinematic viscosity. The purpose of this paper is to show that Burgers vortex is asymptotically stable with respect to general three-dimensional perturbations, for all values of the Reynolds number. This definitive result subsumes earlier studies by various authors, which were either restricted to small Reynolds numbers or to two-dimensional perturbations. Our proof relies on the crucial observation that the linearized operator at Burgers vortex has a simple and very specific dependence upon the axial variable. This allows to reduce the full linearized equations to a vectorial two-dimensional problem, which can be treated using an extension of the techniques developped in earlier works. Although Burgers vortices are found to be stable for all Reynolds numbers, the proof indicates that perturbations may undergo an important transient amplification if $Re$ is large, a phenomenon that was indeed observed in numerical simulations.Comment: 31 pages, no figur

Cosmological Magnetic Fields from Primordial Helicity

Primordial magnetic fields may account for all or part of the fields observed in galaxies. We consider the evolution of the magnetic fields created by pseudoscalar effects in the early universe. Such processes can create force-free fields of maximal helicity; we show that for such a field magnetic energy inverse cascades to larger scales than it would have solely by flux freezing and cosmic expansion. For fields generated at the electroweak phase transition, we find that the predicted wavelength today can in principle be as large as 10 kpc, and the field strength can be as large as 10^{-10} G.Comment: 13 page

Cross helicity and turbulent magnetic diffusivity in the solar convection zone

In a density-stratified turbulent medium the cross helicity is considered as a result of the interaction of the velocity fluctuations and a large-scale magnetic field. By means of a quasilinear theory and by numerical simulations we find the cross helicity and the mean vertical magnetic field anti-correlated. In the high-conductivity limit the ratio of the helicity and the mean magnetic field equals the ratio of the magnetic eddy diffusivity and the (known) density scale height. The result can be used to predict that the cross helicity at the solar surface exceeds the value of 1 Gauss km/s. Its sign is anti-correlated with that of the radial mean magnetic field. Alternatively, we can use our result to determine the value of the turbulent magnetic diffusivity from observations of the cross helicity.Comment: 9 pages, 2 figures, submitted to Solar Physic

Nonlinear stabilitty for steady vortex pairs

In this article, we prove nonlinear orbital stability for steadily translating vortex pairs, a family of nonlinear waves that are exact solutions of the incompressible, two-dimensional Euler equations. We use an adaptation of Kelvin's variational principle, maximizing kinetic energy penalised by a multiple of momentum among mirror-symmetric isovortical rearrangements. This formulation has the advantage that the functional to be maximized and the constraint set are both invariant under the flow of the time-dependent Euler equations, and this observation is used strongly in the analysis. Previous work on existence yields a wide class of examples to which our result applies.Comment: 25 page