97 research outputs found
Bifurcations of blowup in inviscid shell models of convective turbulence
We analyze the blowup (finite-time singularity) in inviscid shell models of
convective turbulence. We show that the blowup exists and its internal
structure undergoes a series of bifurcations under a change of shell model
parameter. Various blowup structures are observed and explained, which vary
from self-similar to periodic, quasi-periodic and chaotic regimes. Though the
blowup takes sophisticated forms, its asymptotic small-scale structure is
independent of initial conditions, i.e., universal. Finally, we discuss
implications of the obtained results for the open problems of blowup in
inviscid flows and for the theory of turbulence.Comment: 22 pages, 10 figure
Stochastic anomaly and large Reynolds number limit in hydrodynamic turbulence models
In this work we address the open problem of high Reynolds number limit in
hydrodynamic turbulence, which we modify by considering a vanishing random
(instead of deterministic) viscosity. In this formulation, a small-scale noise
propagates to large scales in an inverse cascade, which can be described using
qualitative arguments of the Kolmogorov-Obukhov theory. We conjecture that the
limit of the resulting probability distribution exists as , and the limiting flow at finite time remains stochastic even if
forcing, initial and boundary conditions are deterministic. This conjecture is
confirmed numerically for the Sabra model of turbulence, where the solution is
deterministic before and random immediately after a blowup. Then, we derive a
purely inviscid problem formulation with a stochastic boundary condition
imposed in the inertial interval.Comment: 20 pages, 6 figure
Spontaneous stochasticity of velocity in turbulence models
We analyze the phenomenon of spontaneous stochasticity in fluid dynamics
formulated as the nonuniqueness of solutions resulting from viscosity at
infinitesimal scales acting through intermediate on large scales of the flow.
We study the finite-time onset of spontaneous stochasticity in a real version
of the GOY shell model of turbulence. This model allows high-accuracy numerical
simulations for a wide range of scales (up to ten orders of magnitude) and
demonstrates non-chaotic dynamics, but leads to an infinite number of solutions
in the vanishing viscosity limit after the blowup time. Thus, the spontaneous
stochasticity phenomenon is clearly distinguished from the chaotic behavior in
turbulent flows. We provide the numerical and theoretical description of the
system dynamics at all stages. This includes the asymptotic analysis before and
after the blowup leading to universal (periodic and quasi-periodic)
renormalized solutions, followed by nonunique stationary states at large times.Comment: 20 pages, 9 figure
Continuous representation for shell models of turbulence
In this work we construct and analyze continuous hydrodynamic models in one
space dimension, which are induced by shell models of turbulence. After Fourier
transformation, such continuous models split into an infinite number of
uncoupled subsystems, which are all identical to the same shell model. The two
shell models, which allow such a construction, are considered: the dyadic
(Desnyansky--Novikov) model with the intershell ratio and
the Sabra model of turbulence with .
The continuous models allow understanding various properties of shell model
solutions and provide their interpretation in physical space. We show that the
asymptotic solutions of the dyadic model with Kolmogorov scaling correspond to
the shocks (discontinuities) for the induced continuous solutions in physical
space, and the finite-time blowup together with its viscous regularization
follow the scenario similar to the Burgers equation. For the Sabra model, we
provide the physical space representation for blowup solutions and intermittent
turbulent dynamics.Comment: 21 pages, 7 figure
Spontaneously stochastic solutions in one-dimensional inviscid systems
In this paper, we study the inviscid limit of the Sabra shell model of
turbulence, which is considered as a particular case of a viscous conservation
law in one space dimension with a nonlocal quadratic flux function. We present
a theoretical argument (with a detailed numerical confirmation) showing that a
classical deterministic solution before a finite-time blowup, , must
be continued as a stochastic process after the blowup, , representing
a unique physically relevant description in the inviscid limit. This theory is
based on the dynamical system formulation written for the logarithmic time
, which features a stable traveling wave solution for the
inviscid Burgers equation, but a stochastic traveling wave for the Sabra model.
The latter describes a universal onset of stochasticity immediately after the
blowup
Universal structure of blow-up in 1D conservation laws
We discuss universality properties of blow-up of a classical (smooth)
solutions of conservation laws in one space dimension. It is shown that the
renormalized wave profile tends to a universal function, which is independent
both of initial conditions and of the form of a conservation law. This property
is explained in terms of the renormalization group theory. A solitary wave
appears in logarithmic coordinates of the Fourier space as a counterpart of
this universality. Universality is demonstrated in two examples: Burgers
equation and dynamics of ideal polytropic gas.Comment: 8 pages, 2 figure
Blowup as a driving mechanism of turbulence in shell models
Since Kolmogorov proposed his phenomenological theory of hydrodynamic
turbulence in 1941, the description of mechanism leading to the energy cascade
and anomalous scaling remains an open problem in fluid mechanics. Soon after,
in 1949 Onsager noticed that the scaling properties in inertial range imply
non-differentiability of the velocity field in the limit of vanishing
viscosity. This observation suggests that the turbulence mechanism may be
related to a finite-time singularity (blowup) of incompressible Euler
equations. However, the existence of such blowup is still an open problem too.
In this paper, we show that the blowup indeed represents the driving mechanism
of inertial range for a simplified (shell) model of turbulence. Here, blowups
generate coherent structures (instantons), which travel through the inertial
range in finite time and are described by universal self-similar statistics.
The anomaly (deviation of scaling exponents of velocity moments from the
Kolmogorov theory) is related analytically to the process of instanton creation
using the large deviation principle. The results are confirmed by numerical
simulations.Comment: 26 pages, 11 figure
Renormalization group formalism for incompressible Euler equations and the blowup problem
The paper discusses extensions of the renormalization group (RG) formalism
for 3D incompressible Euler equations, which can be used for describing
singularities developing in finite (blowup) or infinite time from smooth
initial conditions of finite energy. In this theory, time evolution is
substituted by the equivalent evolution for renormalized solutions governed by
the RG equations. A fixed point attractor of the RG equations, if it exists,
describes universal self-similar form of observable singularities. This
universality provides a constructive criterion for interpreting results of
numerical experiments. In this paper, renormalization schemes with multiple
spatial scales are developed for the cases of power law and exponential
scaling. The results are compared with the numerical simulations of a
singularity in incompressible Euler equations obtained by Hou and Li (2006) and
Grafke et al. (2008). The comparison supports the conjecture of a singularity
developing exponentially in infinite time and described by a multiple-scale
self-similar asymptotic solution predicted by the RG theory.Comment: 16 pages, 5 figure
Development of high vorticity structures in incompressible 3D Euler equations
We perform the systematic numerical study of high vorticity structures that
develop in the 3D incompressible Euler equations from generic large-scale
initial conditions. We observe that a multitude of high vorticity structures
appear in the form of thin vorticity sheets (pancakes). Our analysis reveals
the self-similarity of the pancakes evolution, which is governed by two
different exponents and describing
compression in the transverse direction and the vorticity growth respectively,
with the universal ratio . We relate
development of these structures to the gradual formation of the Kolmogorov
energy spectrum , which we observe in a fully inviscid
system. With the spectral analysis we demonstrate that the energy transfer to
small scales is performed through the pancake structures, which accumulate in
the Kolmogorov interval of scales and evolve according to the scaling law
for the local vorticity maximums
and the transverse pancake scales .Comment: 31 pages, 18 figure
Light stops at exceptional points
Almost twenty years ago the light was slowed down to less than of
its vacuum speed in a cloud of ultracold atoms of sodium. Upon a sudden
turn-off of the coupling laser, a slow light pulse can be imprinted on cold
atoms such that it can be read out and converted into photon again. In this
process, the light is stopped by absorbing it and storing its shape within the
atomic ensemble. Alternatively, the light can be stopped at the band edge in
photonic-crystal waveguides, where the group speed vanishes. Here we extend the
phenomenon of stopped light to the new field of parity-time (PT) symmetric
systems. We show that zero group speed in PT symmetric optical waveguides can
be achieved if the system is prepared at an exceptional point, where two
optical modes coalesce. This effect can be tuned for optical pulses in a wide
range of frequencies and bandwidths, as we demonstrate in a system of coupled
waveguides with gain and loss.Comment: 5 pages, 4 figure
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