6,194 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
Integrable theories and loop spaces: fundamentals, applications and new developments
We review our proposal to generalize the standard two-dimensional flatness
construction of Lax-Zakharov-Shabat to relativistic field theories in d+1
dimensions. The fundamentals from the theory of connections on loop spaces are
presented and clarified. These ideas are exposed using mathematical tools
familiar to physicists. We exhibit recent and new results that relate the
locality of the loop space curvature to the diffeomorphism invariance of the
loop space holonomy. These result are used to show that the holonomy is abelian
if the holonomy is diffeomorphism invariant.
These results justify in part and set the limitations of the local
implementations of the approach which has been worked out in the last decade.
We highlight very interesting applications like the construction and the
solution of an integrable four dimensional field theory with Hopf solitons, and
new integrability conditions which generalize BPS equations to systems such as
Skyrme theories. Applications of these ideas leading to new constructions are
implemented in theories that admit volume preserving diffeomorphisms of the
target space as symmetries. Applications to physically relevant systems like
Yang Mills theories are summarized. We also discuss other possibilities that
have not yet been explored.Comment: 64 pages, 8 figure
London equation studies of thin-film superconductors with a triangular antidot lattice
We report on a study of vortex pinning in nanoscale antidot defect arrays in
the context of the London Theory. Using a wire network model, we discretize the
array with a fine mesh, thereby providing a detailed treatment of pinning
phenomena. The use of a fine grid has enabled us to examine both circular and
elongated defects, patterned in the form of a rhombus. The latter display
pinning characteristics superior to circular defects constructed with the
similar area. We calculate pinning potentials for defects containing zero and
single quanta, and we obtain a pinning phase diagram for the second matching
field, .Comment: 10 pages and 14 figure
Warped compactification on Abelian vortex in six dimensions
We consider the possibility of localizing gravity on a Nielsen-Olesen vortex
in the context of the Abelian Higgs model. The vortex lives in a
six-dimensional space-time with negative bulk cosmological constant. In this
model we find a region of the parameter space leading, simultaneously, to
warped compactification and to regular space-time geometry. A thin defect limit
is studied. Regular solutions describing warped compactifications in the case
of higher winding number are also presented.Comment: LaTeX, 39 pages, 21 figures, final version appeared in Nucl. Phys.
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