On knotted streamtubes in incompressible hydrodynamical flow and a restricted conserved quantity

For certain families of fluid flow, a new conserved quantity -- stream-helicity -- has been established.Using examples of linked and knotted streamtubes, it has been shown that stream-helicity does, in certain cases, entertain itself with a very precise topological meaning viz, measure of the degree of knottedness or linkage of streamtubes.As a consequence, stream-helicity emerges as a robust topological invariant.Comment: This extended version is the basically a more clarified version of the previous submission physics/0611166v

Optimization of the magnetic dynamo

In stars and planets, magnetic fields are believed to originate from the motion of electrically conducting fluids in their interior, through a process known as the dynamo mechanism. In this Letter, an optimization procedure is used to simultaneously address two fundamental questions of dynamo theory: "Which velocity field leads to the most magnetic energy growth?" and "How large does the velocity need to be relative to magnetic diffusion?" In general, this requires optimization over the full space of continuous solenoidal velocity fields possible within the geometry. Here the case of a periodic box is considered. Measuring the strength of the flow with the root-mean-square amplitude, an optimal velocity field is shown to exist, but without limitation on the strain rate, optimization is prone to divergence. Measuring the flow in terms of its associated dissipation leads to the identification of a single optimal at the critical magnetic Reynolds number necessary for a dynamo. This magnetic Reynolds number is found to be only 15% higher than that necessary for transient growth of the magnetic field.Comment: Optimal velocity field given approximate analytic form. 4 pages, 4 figure

Bipartite partial duals and circuits in medial graphs

It is well known that a plane graph is Eulerian if and only if its geometric dual is bipartite. We extend this result to partial duals of plane graphs. We then characterize all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph.Comment: v2: minor changes. To appear in Combinatoric

Large Scale Structures a Gradient Lines: the case of the Trkal Flow

A specific asymptotic expansion at large Reynolds numbers (R)for the long wavelength perturbation of a non stationary anisotropic helical solution of the force less Navier-Stokes equations (Trkal solutions) is effectively constructed of the Beltrami type terms through multi scaling analysis. The asymptotic procedure is proved to be valid for one specific value of the scaling parameter,namely for the square root of the Reynolds number (R).As a result large scale structures arise as gradient lines of the energy determined by the initial conditions for two anisotropic Beltrami flows of the same helicity.The same intitial conditions determine the boundaries of the vortex-velocity tubes, containing both streamlines and vortex linesComment: 27 pages, 2 figure

Creation and evolution of magnetic helicity

Projecting a non-Abelian SU(2) vacuum gauge field - a pure gauge constructed from the group element U - onto a fixed (electromagnetic) direction in isospace gives rise to a nontrivial magnetic field, with nonvanishing magnetic helicity, which coincides with the winding number of U. Although the helicity is not conserved under Maxwell (vacuum) evolution, it retains one-half its initial value at infinite time.Comment: Clarifying remarks and references added; 12 pages, 1 figure using BoxedEPSF, REVTeX macros; submitted to Phys Rev D; email to [email protected]

Excitation spectroscopy of vortex lattices in a rotating Bose-Einstein condensate

Excitation spectroscopy of vortex lattices in rotating Bose-Einstein condensates is described. We numerically obtain the Bogoliubov-deGenne quasiparticle excitations for a broad range of energies and analyze them in the context of the complex dynamics of the system. Our work is carried out in a regime in which standard hydrodynamic assumptions do not hold, and includes features not readily contained within existing treatments.Comment: 4 pages, 4 figures. Submitted for publicatio

Closure tests for mean field magnetohydrodynamics using a self consistent reduced model

The mean electromotive force and alpha effect are computed for a forced turbulent flow using a simple nonlinear dynamical model. The results are used to check the applicability of two basic analytic ansatze of mean-field magnetohydrodynamics - the second order correlation approximation (SOCA) and the tau approximation. In the numerical simulations the effective Reynolds number Re is 2-20, while the magnetic Prandtl number varies from 0.1 to $10^{7}$. We present evidence that the $\tau$ approximation may be appropriate in dynamical regimes where there is a small-scale dynamo. Catastrophic quenching of the $\alpha$ effect is found for high $P_{m}$. Our results indicate that for high $P_{m}$ SOCA gives a very large value of the $\alpha$ coefficient compared with the ``exact'' solution. The discrepancy depends on the properties of the random force that drives the flow, with a larger difference occuring for $\delta$-correlated force compared with that for a steady random force.Comment: submitted to MNRA

The asymmetry of sunspot cycles and Waldmeier relations as due to nonlinear surface-shear shaped dynamo

The paper presents a study of a solar dynamo model operating in the bulk of the convection zone with the toroidal magnetic field flux concentrated in the subsurface rotational shear layer. We explore how this type of dynamo may depend on spatial variations of turbulent parameters and on the differential rotation near the surface. The mean-field dynamo model takes into account the evolution of magnetic helicity and describes its nonlinear feedback on the generation of large-scale magnetic field by the $\alpha$-effect. We compare the magnetic cycle characteristics predicted by the model, including the cycle asymmetry (associated with the growth and decay times) and the duration - amplitude relation (Waldmeier's effects), with the observed sunspot cycle properties. We show that the model qualitatively reproduces the basic properties of the solar cycles.Comment: 28 pages, 7 figures(Second revision, figures updates

Dynamics of a rolling robot

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