1,068 research outputs found
Geometric phases of water waves
Recently, Banner et al. (2014) highlighted a new fundamental property of open
ocean wave groups, the so-called crest slowdown. For linear narrowband waves,
this is related to the geometric and dynamical phase velocities and
associated with the parallel transport through the principal fiber bundle of
the wave motion with symmetry. The theoretical predictions are
shown to be in fair agreement with ocean field observations, from which the
average crest speed with and
Hamiltonian description and traveling waves of the spatial Dysthe equations
The spatial version of the fourth-order Dysthe equations describe the
evolution of weakly nonlinear narrowband wave trains in deep waters. For
unidirectional waves, the hidden Hamiltonian structure and new invariants are
unveiled by means of a gauge transformation to a new canonical form of the
evolution equations. A highly accurate Fourier-type spectral scheme is
developed to solve for the equations and validate the new conservation laws,
which are satisfied up to machine precision. Further, traveling waves are
numerically investigated using the Petviashvili method. It is found that their
collision appears inelastic, suggesting the non-integrability of the Dysthe
equations.Comment: Research report. 17 pages, 7 figures, 38 references. Other author's
papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh/ . arXiv
admin note: substantial text overlap with arXiv:1110.408
Camassa-Holm type equations for axisymmetric Poiseuille pipe flows
We present a study on the nonlinear dynamics of a disturbance to the laminar
state in non-rotating axisymmetric Poiseuille pipe flows. The associated
Navier-Stokes equations are reduced to a set of coupled generalized
Camassa-Holm type equations. These support singular inviscid travelling waves
with wedge-type singularities, the so called peakons, which bifurcate from
smooth solitary waves as their celerity increase. In physical space they
correspond to localized toroidal vortices or vortexons. The inviscid vortexon
is similar to the nonlinear neutral structures found by Walton (2011) and it
may be a precursor to puffs and slugs observed at transition, since most likely
it is unstable to non-axisymmetric disturbances.Comment: 11 pages, 4 figures, 31 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Kinematics of fluid particles on the sea surface. Hamiltonian theory
We derive the John-Sclavounos equations describing the motion of a fluid
particle on the sea surface from first principles using Lagrangian and
Hamiltonian formalisms applied to the motion of a frictionless particle
constrained on an unsteady surface. The main result is that vorticity generated
on a stress-free surface vanishes at a wave crest when the horizontal particle
velocity equals the crest propagation speed, which is the kinematic criterion
for wave breaking. If this holds for the largest crest, then the symplectic
two-form associated with the Hamiltonian dynamics reduces instantaneously to
that associated with the motion of a particle in free flight, as if the surface
did not exist. Further, exploiting the conservation of the Hamiltonian function
for steady surfaces and traveling waves we show that particle velocities remain
bounded at all times, ruling out the possibility of the finite-time blowup of
solutions
Matrix Geometries Emergent from a Point
We describe a categorical approach to finite noncommutative geometries.
Objects in the category are spectral triples, rather than unitary equivalence
classes as in other approaches. This enables to treat fluctuations of the
metric and unitary equivalences on the same footing, as representatives of
particular morphisms in this category. We then show how a matrix geometry
(Moyal plane) emerges as a fluctuation from one point, and discuss some
geometric aspects of this space.Comment: 1 figur
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