266 research outputs found
Fractal dimension crossovers in turbulent passive scalar signals
The fractal dimension of turbulent passive scalar signals is
calculated from the fluid dynamical equation. depends on the
scale. For small Prandtl (or Schmidt) number one gets two ranges,
for small scale r and =5/3 for large r, both
as expected. But for large one gets a third, intermediate range in
which the signal is extremely wrinkled and has . In that
range the passive scalar structure function has a plateau. We
calculate the -dependence of the crossovers. Comparison with a numerical
reduced wave vector set calculation gives good agreement with our predictions.Comment: 7 pages, Revtex, 3 figures (postscript file on request
Intrinsic Neuronal Properties Switch the Mode of Information Transmission in Networks
Diverse ion channels and their dynamics endow single neurons with complex biophysical properties. These properties determine the heterogeneity of cell types that make up the brain, as constituents of neural circuits tuned to perform highly specific computations. How do biophysical properties of single neurons impact network function? We study a set of biophysical properties that emerge in cortical neurons during the first week of development, eventually allowing these neurons to adaptively scale the gain of their response to the amplitude of the fluctuations they encounter. During the same time period, these same neurons participate in large-scale waves of spontaneously generated electrical activity. We investigate the potential role of experimentally observed changes in intrinsic neuronal properties in determining the ability of cortical networks to propagate waves of activity. We show that such changes can strongly affect the ability of multi-layered feedforward networks to represent and transmit information on multiple timescales. With properties modeled on those observed at early stages of development, neurons are relatively insensitive to rapid fluctuations and tend to fire synchronously in response to wave-like events of large amplitude. Following developmental changes in voltage-dependent conductances, these same neurons become efficient encoders of fast input fluctuations over few layers, but lose the ability to transmit slower, population-wide input variations across many layers. Depending on the neurons' intrinsic properties, noise plays different roles in modulating neuronal input-output curves, which can dramatically impact network transmission. The developmental change in intrinsic properties supports a transformation of a networks function from the propagation of network-wide information to one in which computations are scaled to local activity. This work underscores the significance of simple changes in conductance parameters in governing how neurons represent and propagate information, and suggests a role for background synaptic noise in switching the mode of information transmission
Anomalous Scaling in the N-Point Functions of Passive Scalar
A recent analysis of the 4-point correlation function of the passive scalar
advected by a time-decorrelated random flow is extended to the N-point case. It
is shown that all stationary-state inertial-range correlations are dominated by
homogeneous zero modes of singular operators describing their evolution. We
compute analytically the zero modes governing the N-point structure functions
and the anomalous dimensions corresponding to them to the linear order in the
scaling exponent of the 2-point function of the advecting velocity field. The
implications of these calculations for the dissipation correlations are
discussed.Comment: 16 pages, latex fil
The Viscous Lengths in Hydrodynamic Turbulence are Anomalous Scaling Functions
It is shown that the idea that scaling behavior in turbulence is limited by
one outer length and one inner length is untenable. Every n'th order
correlation function of velocity differences \bbox{\cal
F}_n(\B.R_1,\B.R_2,\dots) exhibits its own cross-over length to
dissipative behavior as a function of, say, . This length depends on
{and on the remaining separations} . One result of this Letter
is that when all these separations are of the same order this length scales
like with
, with being
the scaling exponent of the 'th order structure function. We derive a class
of scaling relations including the ``bridge relation" for the scaling exponent
of dissipation fluctuations .Comment: PRL, Submitted. REVTeX, 4 pages, I fig. (not included) PS Source of
the paper with figure avalable at http://lvov.weizmann.ac.il/onlinelist.htm
A Simple Passive Scalar Advection-Diffusion Model
This paper presents a simple, one-dimensional model of a randomly advected
passive scalar. The model exhibits anomalous inertial range scaling for the
structure functions constructed from scalar differences. The model provides a
simple computational test for recent ideas regarding closure and scaling for
randomly advected passive scalars. Results suggest that high order structure
function scaling depends on the largest velocity eddy size, and hence scaling
exponents may be geometry-dependent and non-universal.Comment: 30 pages, 11 figure
Manifestation of anisotropy persistence in the hierarchies of MHD scaling exponents
The first example of a turbulent system where the failure of the hypothesis
of small-scale isotropy restoration is detectable both in the `flattening' of
the inertial-range scaling exponent hierarchy, and in the behavior of odd-order
dimensionless ratios, e.g., skewness and hyperskewness, is presented.
Specifically, within the kinematic approximation in magnetohydrodynamical
turbulence, we show that for compressible flows, the isotropic contribution to
the scaling of magnetic correlation functions and the first anisotropic ones
may become practically indistinguishable. Moreover, skewness factor now
diverges as the P\'eclet number goes to infinity, a further indication of
small-scale anisotropy.Comment: 4 pages Latex, 1 figur
Renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow
Field theoretical renormalization group methods are applied to a simple model
of a passive scalar quantity advected by the Gaussian non-solenoidal
(``compressible'') velocity field with the covariance . Convective range anomalous scaling for the structure
functions and various pair correlators is established, and the corresponding
anomalous exponents are calculated to the order of the
expansion. These exponents are non-universal, as a result of the degeneracy of
the RG fixed point. In contrast to the case of a purely solenoidal velocity
field (Obukhov--Kraichnan model), the correlation functions in the case at hand
exhibit nontrivial dependence on both the IR and UV characteristic scales, and
the anomalous scaling appears already at the level of the pair correlator. The
powers of the scalar field without derivatives, whose critical dimensions
determine the anomalous exponents, exhibit multifractal behaviour. The exact
solution for the pair correlator is obtained; it is in agreement with the
result obtained within the expansion. The anomalous exponents for
passively advected magnetic fields are also presented in the first order of the
expansion.Comment: 31 pages, REVTEX file. More detailed discussion of the
one-dimensional case and comparison to the previous paper [20] are given;
references updated. Results and formulas unchange
Anomalous scaling in random shell models for passive scalars
A shell-model version of Kraichnan's (1994 {\it Phys. Rev. Lett. \bf 72},
1016) passive scalar problem is introduced which is inspired from the model of
Jensen, Paladin and Vulpiani (1992 {\it Phys. Rev. A\bf 45}, 7214). As in the
original problem, the prescribed random velocity field is Gaussian,
delta-correlated in time and has a power-law spectrum ,
where is the wavenumber. Deterministic differential equations for second
and fourth-order moments are obtained and then solved numerically. The
second-order structure function of the passive scalar has normal scaling, while
the fourth-order structure function has anomalous scaling. For the
anomalous scaling exponents are determined for structure functions up
to by Monte Carlo simulations of the random shell model, using a
stochastic differential equation scheme, validated by comparison with the
results obtained for the second and fourth-order structure functions.Comment: Plain LaTex, 15 pages, 4 figure available upon request to
[email protected]
Exact Resummations in the Theory of Hydrodynamic Turbulence: III. Scenarios for Anomalous Scaling and Intermittency
Elements of the analytic structure of anomalous scaling and intermittency in
fully developed hydrodynamic turbulence are described. We focus here on the
structure functions of velocity differences that satisfy inertial range scaling
laws , and the correlation of energy dissipation
. The goal is to understand the
exponents and from first principles. In paper II of this series
it was shown that the existence of an ultraviolet scale (the dissipation scale
) is associated with a spectrum of anomalous exponents that characterize
the ultraviolet divergences of correlations of gradient fields. The leading
scaling exponent in this family was denoted . The exact resummation of
ladder diagrams resulted in the calculation of which satisfies the
scaling relation . In this paper we continue our analysis and
show that nonperturbative effects may introduce multiscaling (i.e.
not being linear in ) with the renormalization scale being the infrared
outer scale of turbulence . It is shown that deviations from K41 scaling of
() must appear if the correlation of dissipation is
mixing (i.e. ). We derive an exact scaling relation . We present analytic expressions for for all
and discuss their relation to experimental data. One surprising prediction is
that the time decay constant of scales
independently of : the dynamic scaling exponent is the same for all
-order quantities, .Comment: PRE submitted, 22 pages + 11 figures, REVTeX. The Eps files of
figures will be FTPed by request to [email protected]
Towards a Nonperturbative Theory of Hydrodynamic Turbulence:Fusion Rules, Exact Bridge Relations and Anomalous Viscous Scaling Functions
In this paper we derive here, on the basis of the NS eqs. a set of fusion
rules for correlations of velocity differences when all the separation are in
the inertial interval. Using this we consider the standard hierarchy of
equations relating the -th order correlations (originating from the viscous
term in the NS eq.) to 'th order (originating from the nonlinear term) and
demonstrate that for fully unfused correlations the viscous term is negligible.
Consequently the hierarchic chain is decoupled in the sense that the
correlations of 'th order satisfy a homogeneous equation that may exhibit
anomalous scaling solutions. Using the same hierarchy of eqs. when some
separations go to zero we derive a second set of fusion rules for correlations
with differences in the viscous range. The latter includes gradient fields. We
demonstrate that every n'th order correlation function of velocity differences
{\cal F}_n(\B.R_1,\B.R_2,\dots) exhibits its own cross-over length
to dissipative behavior as a function of, say, . This length depends on
{and on the remaining separations} . When all these
separations are of the same order this length scales like with ,
with being the scaling exponent of the 'th order structure
function. We derive a class of exact scaling relations bridging the exponents
of correlations of gradient fields to the exponents of the 'th
order structure functions. One of these relations is the well known ``bridge
relation" for the scaling exponent of dissipation fluctuations .Comment: PRE, Submitted. REVTeX, 18 pages, 7 figures (not included) PS Source
of the paper with figures avalable at
http://lvov.weizmann.ac.il/onlinelist.htm
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