27,725 research outputs found
On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions
For a strictly inviscid barotropic flow with conservative body forces, the Helmholtz vorticity theorem shows that material or Lagrangian surfaces which are vortex surfaces at time t = 0 remain so for t > 0. In this study, a systematic methodology is developed for constructing smooth scalar fields φ(x, y, z, t = 0) for Taylor–Green and Kida–Pelz velocity fields, which, at t = 0, satisfy ω·∇φ = 0. We refer to such fields as vortex-surface fields. Then, for some constant C, iso-surfaces φ = C define vortex surfaces. It is shown that, given the vorticity, our definition of a vortex-surface field admits non-uniqueness, and this is presently resolved numerically using an optimization approach. Additionally, relations between vortex-surface fields and the classical Clebsch representation are discussed for flows with zero helicity. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Both uniqueness and the distinction separating the evolution of vortex-surface fields and Lagrangian fields are discussed. By tracking φ as a Lagrangian field in slightly viscous flows, we show that the well-defined evolution of Lagrangian surfaces that are initially vortex surfaces can be a good approximation to vortex surfaces at later times prior to vortex reconnection. In the evolution of such Lagrangian fields, we observe that initially blob-like vortex surfaces are progressively stretched to sheet-like shapes so that neighbouring portions approach each other, with subsequent rolling up of structures near the interface, which reveals more information on dynamics than the iso-surfaces of vorticity magnitude. The non-local geometry in the evolution is quantified by two differential geometry properties. Rolled-up local shapes are found in the Lagrangian structures that were initially vortex surfaces close to the time of vortex reconnection. It is hypothesized that this is related to the formation of the very high vorticity regions
Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces
We establish the background for the study of geodesics on noncompact
polygonal surfaces. For illustration, we study the recurrence of geodesics on
-periodic polygonal surfaces. We prove, in particular, that almost all
geodesics on a topologically typical -periodic surface with boundary are
recurrent.Comment: 34 pages, 13 figures. To be published in V. V. Kozlov's Festschrif
Universal circles for quasigeodesic flows
We show that if M is a hyperbolic 3-manifold which admits a quasigeodesic
flow, then pi_1(M) acts faithfully on a universal circle by homeomorphisms, and
preserves a pair of invariant laminations of this circle. As a corollary, we
show that the Thurston norm can be characterized by quasigeodesic flows,
thereby generalizing a theorem of Mosher, and we give the first example of a
closed hyperbolic 3-manifold without a quasigeodesic flow, answering a
long-standing question of Thurston.Comment: This is the version published by Geometry & Topology on 29 November
2006. V4: typsetting correction
Multitoroidal configurations as equilibrium flow eigenstates
Equilibrium eigenstates of an axisymmetric magnetically confined plasma with
toroidal flow are investigated by means of exact solutions of the ideal
magnetohydrodynamic equations. The study includes "compressible" flows with
constant temperature, but varying density on magnetic surfaces and
incompressible ones with constant density, but varying temperature thereon. In
both cases eigenfunctions of the form Psi_{nl} = Z_l(z)R_n(R) (l, n=1,2,...)
describe configurations with lxn magnetic axes. By varying the flow parameters
a change in magnetic topology is possible. In addition, the effects of the flow
and the aspect ratio on the Shafranov shift are evaluated along with the
variations of density and temperature on magnetic surfaces.Comment: 12th International Congress on Plasma Physics, 25-29 October 2004,
Nice (France
A Spinning Anti-de Sitter Wormhole
We construct a 2+1 dimensional spacetime of constant curvature whose spatial
topology is that of a torus with one asymptotic region attached. It is also a
black hole whose event horizon spins with respect to infinity. An observer
entering the hole necessarily ends up at a "singularity"; there are no inner
horizons.
In the construction we take the quotient of 2+1 dimensional anti-de Sitter
space by a discrete group Gamma. A key part of the analysis proceeds by
studying the action of Gamma on the boundary of the spacetime.Comment: Latex, 28 pages, 7 postscript figures included in text, a Latex file
without figures can be found at http://vanosf.physto.se/~stefan/spinning.html
Replaced with journal version, minor change
Combinatorial Calabi flows on surfaces
For triangulated surfaces, we introduce the combinatorial Calabi flow which
is an analogue of smooth Calabi flow. We prove that the solution of
combinatorial Calabi flow exists for all time. Moreover, the solution converges
if and only if Thurston's circle packing exists. As a consequence,
combinatorial Calabi flow provides a new algorithm to find circle packings with
prescribed curvatures. The proofs rely on careful analysis of combinatorial
Calabi energy, combinatorial Ricci potential and discrete dual-Laplacians.Comment: 17 pages, 5 figure
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