119 research outputs found
The global picture of self-similar and not self-similar decay in Burgers Turbulence
This paper continue earlier investigations on the decay of Burgers turbulence
in one dimension from Gaussian random initial conditions of the power-law
spectral type . Depending on the power , different
characteristic regions are distinguished. The main focus of this paper is to
delineate the regions in wave-number and time in which self-similarity
can (and cannot) be observed, taking into account small- and large-
cutoffs. The evolution of the spectrum can be inferred using physical arguments
describing the competition between the initial spectrum and the new frequencies
generated by the dynamics. For large wavenumbers, we always have
region, associated to the shocks. When is less than one, the large-scale
part of the spectrum is preserved in time and the global evolution is
self-similar, so that scaling arguments perfectly predict the behavior in time
of the energy and of the integral scale. If is larger than two, the
spectrum tends for long times to a universal scaling form independent of the
initial conditions, with universal behavior at small wavenumbers. In the
interval the leading behaviour is self-similar, independent of and
with universal behavior at small wavenumber. When , the spectrum
has three scaling regions : first, a region at very small \ms1 with
a time-independent constant, second, a region at intermediate
wavenumbers, finally, the usual region. In the remaining interval,
the small- cutoff dominates, and also plays no role. We find also
(numerically) the subleading term in the evolution of the spectrum
in the interval . High-resolution numerical simulations have been
performed confirming both scaling predictions and analytical asymptotic theory.Comment: 14 pages, 19 figure
On the decay of Burgers turbulence
This work is devoted to the decay ofrandom solutions of the unforced Burgers
equation in one dimension in the limit of vanishing viscosity. The initial
velocity is homogeneous and Gaussian with a spectrum proportional to at
small wavenumbers and falling off quickly at large wavenumbers. In physical
space, at sufficiently large distances, there is an ``outer region'', where the
velocity correlation function preserves exactly its initial form (a power law)
when is not an even integer. When the spectrum, at long times, has
three scaling regions : first, a region at very small \ms1 with a
time-independent constant, stemming from this outer region, in which the
initial conditions are essentially frozen; second, a region at
intermediate wavenumbers, related to a self-similarly evolving ``inner region''
in physical space and, finally, the usual region, associated to the
shocks. The switching from the to the region occurs around a wave
number , while the switching from to
occurs around (ignoring logarithmic
corrections in both instances). The key element in the derivation of the
results is an extension of the Kida (1979) log-corrected law for the
energy decay when to the case of arbitrary integer or non-integer .
A systematic derivation is given in which both the leading term and estimates
of higher order corrections can be obtained. High-resolution numerical
simulations are presented which support our findings.Comment: In LaTeX with 11 PostScript figures. 56 pages. One figure contributed
by Alain Noullez (Observatoire de Nice, France
Instanton Theory of Burgers Shocks and Intermittency
A lagrangian approach to Burgers turbulence is carried out along the lines of
the field theoretical Martin-Siggia-Rose formalism of stochastic hydrodynamics.
We derive, from an analysis based on the hypothesis of unbroken galilean
invariance, the asymptotic form of the probability distribution function of
negative velocity-differences. The origin of Burgers intermittency is found to
rely on the dynamical coupling between shocks, identified to instantons, and
non-coherent background fluctuations, which, then, cannot be discarded in a
consistent statistical description of the flow.Comment: 7 pages; LaTe
Cosmological Perturbation Theory Using the Schr\"odinger Equation
We introduce the theory of non-linear cosmological perturbations using the
correspondence limit of the Schr\"odinger equation. The resulting formalism is
equivalent to using the collisionless Boltzman (or Vlasov) equations which
remain valid during the whole evolution, even after shell crossing. Other
formulations of perturbation theory explicitly break down at shell crossing,
e.g. Eulerean perturbation theory, which describes gravitational collapse in
the fluid limit. This paper lays the groundwork by introducing the new
formalism, calculating the perturbation theory kernels which form the basis of
all subsequent calculations. We also establish the connection with conventional
perturbation theories, by showing that third order tree level results, such as
bispectrum, skewness, cumulant correlators, three-point function are exactly
reproduced in the appropriate expansion of our results. We explicitly show that
cumulants up to N=5 predicted by Eulerian perturbation theory for the dark
matter field are exactly recovered in the corresponding limit. A
logarithmic mapping of the field naturally arises in the Schr\"odinger context,
which means that tree level perturbation theory translates into (possibly
incomplete) loop corrections for the conventional perturbation theory. We show
that the first loop correction for the variance is for a field with spectral index . This yields 1.86 and
0.86 for respectively, and to be compared with the exact loop order
corrections 1.82, and 0.88. Thus our tree-level theory recovers the dominant
part of first order loop corrections of the conventional theory, while
including (partial) loop corrections to infinite order in terms of .Comment: 5 pages, submitted to ApJ Letter
Is the cosmic UV background fluctuating at redshift z ~ 6 ?
We study the Gunn-Peterson effect of the photo-ionized intergalactic
medium(IGM) in the redshift range 5< z <6.4 using semi-analytic simulations
based on the lognormal model. Assuming a rapidly evolved and spatially uniform
ionizing background, the simulation can produce all the observed abnormal
statistical features near redshift z ~ 6. They include: 1) rapidly increase of
absorption depths; 2) large scatter in the optical depths; 3) long-tailed
distributions of transmitted flux and 4) long dark gaps in spectra. These
abnormal features are mainly due to rare events, which correspond to the
long-tailed probability distribution of the IGM density field, and therefore,
they may not imply significantly spatial fluctuations in the UV ionizing
background at z ~ 6.Comment: 12 pages, 4 figs, accepted by ApJ
An excursion set model of the cosmic web: The abundance of sheets, filaments and halos
We discuss an analytic approach for modeling structure formation in sheets,
filaments and knots. This is accomplished by combining models of triaxial
collapse with the excursion set approach: sheets are defined as objects which
have collapsed along only one axis, filaments have collapsed along two axes,
and halos are objects in which triaxial collapse is complete. In the simplest
version of this approach, which we develop here, large scale structure shows a
clear hierarchy of morphologies: the mass in large-scale sheets is partitioned
up among lower mass filaments, which themselves are made-up of still lower mass
halos. Our approach provides analytic estimates of the mass fraction in sheets,
filaments and halos, and its evolution, for any background cosmological model
and any initial fluctuation spectrum. In the currently popular CDM
model, our analysis suggests that more than 99% of the cosmic mass is in
sheets, and 72% in filaments, with mass larger than at the
present time. For halos, this number is only 46%. Our approach also provides
analytic estimates of how halo abundances at any given time correlate with the
morphology of the surrounding large-scale structure, and how halo evolution
correlates with the morphology of large scale structure.Comment: 22 pages, 7 figures, Accepted for publication in Ap
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