12,452 research outputs found
A Nonlinear Coupling Network to Simulate the Development of the r-mode Instablility in Neutron Stars II. Dynamics
Two mechanisms for nonlinear mode saturation of the r-mode in neutron stars
have been suggested: the parametric instability mechanism involving a small
number of modes and the formation of a nearly continuous Kolmogorov-type
cascade. Using a network of oscillators constructed from the eigenmodes of a
perfect fluid incompressible star, we investigate the transition between the
two regimes numerically. Our network includes the 4995 inertial modes up to n<=
30 with 146,998 direct couplings to the r-mode and 1,306,999 couplings with
detuning< 0.002 (out of a total of approximately 10^9 possible couplings).
The lowest parametric instability thresholds for a range of temperatures are
calculated and it is found that the r-mode becomes unstable to modes with
13<n<15. In the undriven, undamped, Hamiltonian version of the network the rate
to achieve equipartition is found to be amplitude dependent, reminiscent of the
Fermi-Pasta-Ulam problem. More realistic models driven unstable by
gravitational radiation and damped by shear viscosity are explored next. A
range of damping rates, corresponding to temperatures 10^6K to 10^9K, is
considered. Exponential growth of the r-mode is found to cease at small
amplitudes, approximately 10^-4. For strongly damped, low temperature models, a
few modes dominate the dynamics. The behavior of the r-mode is complicated, but
its amplitude is still no larger than about 10^-4 on average. For high
temperature, weakly damped models the r-mode feeds energy into a sea of
oscillators that achieve approximate equipartition. In this case the r-mode
amplitude settles to a value for which the rate to achieve equipartition is
approximately the linear instability growth rate.Comment: 18 Pages 14 Figure
Observers for compressible Navier-Stokes equation
We consider a multi-dimensional model of a compressible fluid in a bounded
domain. We want to estimate the density and velocity of the fluid, based on the
observations for only velocity. We build an observer exploiting the symmetries
of the fluid dynamics laws. Our main result is that for the linearised system
with full observations of the velocity field, we can find an observer which
converges to the true state of the system at any desired convergence rate for
finitely many but arbitrarily large number of Fourier modes. Our
one-dimensional numerical results corroborate the results for the linearised,
fully observed system, and also show similar convergence for the full nonlinear
system and also for the case when the velocity field is observed only over a
subdomain
Nonlinear modes of clarinet-like musical instruments
The concept of nonlinear modes is applied in order to analyze the behavior of
a model of woodwind reed instruments. Using a modal expansion of the impedance
of the instrument, and by projecting the equation for the acoustic pressure on
the normal modes of the air column, a system of second order ordinary
differential equations is obtained. The equations are coupled through the
nonlinear relation describing the volume flow of air through the reed channel
in response to the pressure difference across the reed. The system is treated
using an amplitude-phase formulation for nonlinear modes, where the frequency
and damping functions, as well as the invariant manifolds in the phase space,
are unknowns to be determined. The formulation gives, without explicit
integration of the underlying ordinary differential equation, access to the
transient, the limit cycle, its period and stability. The process is
illustrated for a model reduced to three normal modes of the air column
Gravitational Wavetrains in the Quasi-Equilibrium Approximation: A Model Problem in Scalar Gravitation
A quasi-equilibrium (QE) computational scheme was recently developed in
general relativity to calculate the complete gravitational wavetrain emitted
during the inspiral phase of compact binaries. The QE method exploits the fact
that the the gravitational radiation inspiral timescale is much longer than the
orbital period everywhere outside the ISCO. Here we demonstrate the validity
and advantages of the QE scheme by solving a model problem in relativistic
scalar gravitation theory. By adopting scalar gravitation, we are able to
numerically track without approximation the damping of a simple, quasi-periodic
radiating system (an oscillating spherical matter shell) to final equilibrium,
and then use the exact numerical results to calibrate the QE approximation
method. In particular, we calculate the emitted gravitational wavetrain three
different ways: by integrating the exact coupled dynamical field and matter
equations, by using the scalar-wave monopole approximation formula
(corresponding to the quadrupole formula in general relativity), and by
adopting the QE scheme. We find that the monopole formula works well for weak
field cases, but fails when the fields become even moderately strong. By
contrast, the QE scheme remains quite reliable for moderately strong fields,
and begins to breakdown only for ultra-strong fields. The QE scheme thus
provides a promising technique to construct the complete wavetrain from binary
inspiral outside the ISCO, where the gravitational fields are strong, but where
the computational resources required to follow the system for more than a few
orbits by direct numerical integration of the exact equations are prohibitive.Comment: 15 pages, 14 figure
Bessel Functions in Mass Action. Modeling of Memories and Remembrances
Data from experimental observations of a class of neurological processes
(Freeman K-sets) present functional distribution reproducing Bessel function
behavior. We model such processes with couples of damped/amplified oscillators
which provide time dependent representation of Bessel equation. The root loci
of poles and zeros conform to solutions of K-sets. Some light is shed on the
problem of filling the gap between the cellular level dynamics and the brain
functional activity. Breakdown of time-reversal symmetry is related with the
cortex thermodynamic features. This provides a possible mechanism to deduce
lifetime of recorded memory.Comment: 16 pages, 9 figures, Physics Letters A, 2015 in pres
Synchronization framework for modeling transition to thermoacoustic instability in laminar combustors
We, herein, present a new model based on the framework of synchronization to
describe a thermoacoustic system and capture the multiple bifurcations that
such a system undergoes. Instead of applying flame describing function to
depict the unsteady heat release rate as the flame's response to acoustic
perturbation, the new model considers the acoustic field and the unsteady heat
release rate as a pair of nonlinearly coupled damped oscillators. By varying
the coupling strength, multiple dynamical behaviors, including limit cycle
oscillation, quasi-periodic oscillation, strange nonchaos, and chaos can be
captured. Furthermore, the model was able to qualitatively replicate the
different behaviors of a laminar thermoacoustic system observed in experiments
by Kabiraj et al.~[Chaos 22, 023129 (2012)]. By analyzing the temporal
variation of the phase difference between heat release rate oscillations and
pressure oscillations under different dynamical states, we show that the
characteristics of the dynamical states depend on the nature of synchronization
between the two signals, which is consistent with previous experimental
findings.Comment: 18 pages, 7 figure
Dynamics of Lattice Kinks
In this paper we consider two models of soliton dynamics (the sine Gordon and
the \phi^4 equations) on a 1-dimensional lattice. We are interested in
particular in the behavior of their kink-like solutions inside the Peierls-
Nabarro barrier and its variation as a function of the discreteness parameter.
We find explicitly the asymptotic states of the system for any value of the
discreteness parameter and the rates of decay of the initial data to these
asymptotic states. We show that genuinely periodic solutions are possible and
we identify the regimes of the discreteness parameter for which they are
expected to persist. We also prove that quasiperiodic solutions cannot exist.
Our results are verified by numerical simulations.Comment: 50 pages, 10 figures, LaTeX documen
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