472 research outputs found
An early warning indicator for atmospheric blocking events using transfer operators
The existence of persistent midlatitude atmospheric flow regimes with
time-scales larger than 5-10 days and indications of preferred transitions
between them motivates to develop early warning indicators for such regime
transitions. In this paper, we use a hemispheric barotropic model together with
estimates of transfer operators on a reduced phase space to develop an early
warning indicator of the zonal to blocked flow transition in this model. It is
shown that, the spectrum of the transfer operators can be used to study the
slow dynamics of the flow as well as the non-Markovian character of the
reduction. The slowest motions are thereby found to have time scales of three
to six weeks and to be associated with meta-stable regimes (and their
transitions) which can be detected as almost-invariant sets of the transfer
operator. From the energy budget of the model, we are able to explain the
meta-stability of the regimes and the existence of preferred transition paths.
Even though the model is highly simplified, the skill of the early warning
indicator is promising, suggesting that the transfer operator approach can be
used in parallel to an operational deterministic model for stochastic
prediction or to assess forecast uncertainty
Hyperacceleration in a stochastic Fermi-Ulam model
Fermi acceleration in a Fermi-Ulam model, consisting of an ensemble of
particles bouncing between two, infinitely heavy, stochastically oscillating
hard walls, is investigated. It is shown that the widely used approximation,
neglecting the displacement of the walls (static wall approximation), leads to
a systematic underestimation of particle acceleration. An improved
approximative map is introduced, which takes into account the effect of the
wall displacement, and in addition allows the analytical estimation of the long
term behavior of the particle mean velocity as well as the corresponding
probability distribution, in complete agreement with the numerical results of
the exact dynamics. This effect accounting for the increased particle
acceleration -Fermi hyperacceleration- is also present in higher dimensional
systems, such as the driven Lorentz gas.Comment: 4 pages, 3 figures. To be published in Phys. Rev. Let
Optimally coherent sets in geophysical flows: A new approach to delimiting the stratospheric polar vortex
The "edge" of the Antarctic polar vortex is known to behave as a barrier to
the meridional (poleward) transport of ozone during the austral winter. This
chemical isolation of the polar vortex from the middle and low latitudes
produces an ozone minimum in the vortex region, intensifying the ozone hole
relative to that which would be produced by photochemical processes alone.
Observational determination of the vortex edge remains an active field of
research. In this letter, we obtain objective estimates of the structure of the
polar vortex by introducing a new technique based on transfer operators that
aims to find regions with minimal external transport. Applying this new
technique to European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-40
three-dimensional velocity data we produce an improved three-dimensional
estimate of the vortex location in the upper stratosphere where the vortex is
most pronounced. This novel computational approach has wide potential
application in detecting and analysing mixing structures in a variety of
atmospheric, oceanographic, and general fluid dynamical settings
Spectral degeneracy and escape dynamics for intermittent maps with a hole
We study intermittent maps from the point of view of metastability. Small
neighbourhoods of an intermittent fixed point and their complements form pairs
of almost-invariant sets. Treating the small neighbourhood as a hole, we first
show that the absolutely continuous conditional invariant measures (ACCIMs)
converge to the ACIM as the length of the small neighbourhood shrinks to zero.
We then quantify how the escape dynamics from these almost-invariant sets are
connected with the second eigenfunctions of Perron-Frobenius (transfer)
operators when a small perturbation is applied near the intermittent fixed
point. In particular, we describe precisely the scaling of the second
eigenvalue with the perturbation size, provide upper and lower bounds, and
demonstrate convergence of the positive part of the second eigenfunction
to the ACIM as the perturbation goes to zero. This perturbation and associated
eigenvalue scalings and convergence results are all compatible with Ulam's
method and provide a formal explanation for the numerical behaviour of Ulam's
method in this nonuniformly hyperbolic setting. The main results of the paper
are illustrated with numerical computations.Comment: 34 page
An Infinite Swapping Approach to the Rare-Event Sampling Problem
We describe a new approach to the rare-event Monte Carlo sampling problem.
This technique utilizes a symmetrization strategy to create probability
distributions that are more highly connected and thus more easily sampled than
their original, potentially sparse counterparts. After discussing the formal
outline of the approach and devising techniques for its practical
implementation, we illustrate the utility of the technique with a series of
numerical applications to Lennard-Jones clusters of varying complexity and
rare-event character.Comment: 24 pages, 16 figure
Expected length of the longest common subsequence for large alphabets
We consider the length L of the longest common subsequence of two randomly
uniformly and independently chosen n character words over a k-ary alphabet.
Subadditivity arguments yield that the expected value of L, when normalized by
n, converges to a constant C_k. We prove a conjecture of Sankoff and Mainville
from the early 80's claiming that C_k\sqrt{k} goes to 2 as k goes to infinity.Comment: 14 pages, 1 figure, LaTe
Perturbation of an Eigen-Value from a Dense Point Spectrum : An Example
We study a perturbed Floquet Hamiltonian depending on a coupling
constant . The spectrum is assumed to be pure point and
dense. We pick up an eigen-value, namely , and show the
existence of a function defined on such that
for all , 0 is a point of
density for the set , and the Rayleigh-Schr\"odinger perturbation series
represents an asymptotic series for the function . All ideas
are developed and demonstrated when treating an explicit example but some of
them are expected to have an essentially wider range of application.Comment: Latex, 24 pages, 51
Bifurcations of discrete breathers in a diatomic Fermi-Pasta-Ulam chain
Discrete breathers are time-periodic, spatially localized solutions of the
equations of motion for a system of classical degrees of freedom interacting on
a lattice. Such solutions are investigated for a diatomic Fermi-Pasta-Ulam
chain, i. e., a chain of alternate heavy and light masses coupled by anharmonic
forces. For hard interaction potentials, discrete breathers in this model are
known to exist either as ``optic breathers'' with frequencies above the optic
band, or as ``acoustic breathers'' with frequencies in the gap between the
acoustic and the optic band. In this paper, bifurcations between different
types of discrete breathers are found numerically, with the mass ratio m and
the breather frequency omega as bifurcation parameters. We identify a period
tripling bifurcation around optic breathers, which leads to new breather
solutions with frequencies in the gap, and a second local bifurcation around
acoustic breathers. These results provide new breather solutions of the FPU
system which interpolate between the classical acoustic and optic modes. The
two bifurcation lines originate from a particular ``corner'' in parameter space
(omega,m). As parameters lie near this corner, we prove by means of a center
manifold reduction that small amplitude solutions can be described by a
four-dimensional reversible map. This allows us to derive formally a continuum
limit differential equation which characterizes at leading order the
numerically observed bifurcations.Comment: 30 pages, 10 figure
How well can one resolve the state space of a chaotic map?
All physical systems are affected by some noise that limits the resolution
that can be attained in partitioning their state space. For chaotic, locally
hyperbolic flows, this resolution depends on the interplay of the local
stretching/contraction and the smearing due to noise. We propose to determine
the `finest attainable' partition for a given hyperbolic dynamical system and a
given weak additive white noise, by computing the local eigenfunctions of the
adjoint Fokker-Planck operator along each periodic point, and using overlaps of
their widths as the criterion for an optimal partition. The Fokker-Planck
evolution is then represented by a finite transition graph, whose spectral
determinant yields time averages of dynamical observables. Numerical tests of
such `optimal partition' of a one-dimensional repeller support our hypothesis.Comment: 4 pages, 3 postscript figures, uses revtex4; changed conten
Penetration of hot electrons through a cold disordered wire
We study a penetration of an electron with high energy E<<T through strongly
disordered wire of length L<<a (a being the localization length). Such an
electron can loose, but not gain the energy, when hopping from one localized
state to another. We have found a distribution function for the transmission
coefficient t. The typical t remains exponentially small in L/a, but with the
decrement, reduced compared to the case of direct elastic tunnelling. The
distribution function has a relatively strong tail in the domain of anomalously
high t; the average ~(a/L)^2 is controlled by rare configurations of
disorder, corresponding to this tail.Comment: 4 pages, 5 figure
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