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| Universal Profile of the Vortex Condensate in Two-Dimensional Turbulence
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| Jason Laurie, Guido Boffetta, Gregory Falkovich, Igor Kolokolov, and Vladimir Lebedev
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Physical Review Letters 113, 254503 (2014).
An inverse turbulent cascade in a restricted two-dimensional periodic domain
creates a condensate - a pair of coherent system-size vortices. We perform
extensive numerical simulations of this system and carry out theoretical
analysis based on momentum and energy exchanges between the turbulence and the
vortices. We show that the vortices have a universal internal structure
independent of the type of small-scale dissipation, small-scale forcing, and
boundary conditions. The theory predicts not only the vortex inner region
profile, but also the amplitude, which both perfectly agree with the numerical
data.
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| Gyrotactic trapping in laminar and turbulent Kolmogorov flow
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| Francesco Santamaria, Filippo De Lillo, Massimo Cencini and Guido Boffetta
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Physics of Fluids 26, 111901 (2014).
Phytoplankton patchiness, namely the heterogeneous distribution of microalgae
over multiple spatial scales, dramatically impacts marine ecology. A
spectacular example of such heterogeneity occurs in thin phytoplankton layers
(TPLs), where large numbers of photosynthetic microorganisms are found within a
small depth interval. Some species of motile phytoplankton can form TPLs by
gyrotactic trapping due to the interplay of their particular swimming style
(directed motion biased against gravity) and the transport by a flow with shear
along the direction of gravity. Here we consider gyrotactic swimmers in
numerical simulations of the Kolmogorov shear flow, both in laminar and
turbulent regimes. In the laminar case, we show that the swimmer motion is
integrable and the formation of TPLs can be fully characterized by means of
dynamical systems tools. We then study the effects of rotational Brownian
motion or turbulent fluctuations (appearing when the Reynolds number is large
enough) on TPLs. In both cases, we show that TPLs become transient, and we
characterize their persistence.
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| Redistribution of Kinetic Energy in Turbulent Flows
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| Alain Pumir, Haitao Xu, Guido Boffetta, Gregory Falkovich
and Eberhard Bodenschatz
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Physical Review X 4, 041006 (2014).
In statistically homogeneous turbulent flows, pressure forces provide the main
mechanism to redistribute kinetic energy among fluid elements, without net
contribution to the overall energy budget. This holds true in both
two-dimensional (2D) and three-dimensional (3D) flows, which show fundamentally
different physics. As we demonstrate here, pressure forces act on fluid
elements very differently in these two cases. We find in numerical simulations
that in 3D pressure forces strongly accelerate the fastest fluid elements, and
that in 2D this effect is absent. In 3D turbulence, our findings put forward a
mechanism for a possibly singular buildup of energy, and thus may shed new
light on the smoothness problem of the solution of the Navier-Stokes equation
in 3D.
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| Dimensional transition in rotating turbulence
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| E. Deusebio, G. Boffetta, E. Lidborg and S. Musacchio
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Physical Review E 90, 023005 (2014).
In this work we investigate, by means of direct numerical hyperviscous
simulations, how rotation affects the bidimensionalization of a turbulent flow.
We study a thin layer of fluid, forced by a two-dimensional forcing, within the
framework of the "split cascade" in which the injected energy flows
both to small scales (generating the direct cascade) and to large scale (to
form the inverse cascade). It is shown that rotation reinforces the inverse
cascade at the expense of the direct one, thus promoting bidimensionalization
of the flow. This is achieved by a suppression of the enstrophy production at
large scales. Nonetheless, we find that, in the range of rotation rates
investigated, increasing the vertical size of the computational domain causes a
reduction of the flux of the inverse cascade. Our results suggest that, even in
rotating flows, the inverse cascade may eventually disappear when the vertical
scale is sufficiently large with respect to the forcing scale. We also study
how the split cascade and confinement influence the breaking of symmetry
induced by rotation.
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| Flight-crash events in turbulence
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| Haitao Xu, Alain Pumir, Gregory Falkovich, Eberhard Bodenschatz,
Michael Shats, Hua Xia, Nicolas Francois and Guido Boffetta
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PNAS 1321682111 (2014).
Irreversibility is a fundamental aspect of the evolution of natural systems,
and quantifying its manifestations is a challenge in any attempt to describe
nonequilibrium systems. In the case of fluid turbulence, an emblematic example
of a system very far from equilibrium, we show that the motion of a single
fluid particle provides a clear manifestation of time irreversibility. Namely,
we observe that fluid particles tend to lose kinetic energy faster than they
gain it. This is best seen by the presence of rare "flight-crash" events, where
fast moving particles suddenly decelerate into a region where fluid motion is
slow. Remarkably, the statistical signature of these events establishes a
quantitative relation between the degree of irreversibility and turbulence
intensity.
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| Turbulent channel without boundaries: The periodic Kolmogorov flow
|
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| S. Musacchio and G. Boffetta
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Physical Review E 89, 023004 (2014).
The Kolmogorov flow provides an ideal instance of a virtual channel flow: It
has no boundaries, but it possesses well defined mean flow in each half
wavelength. We exploit this remarkable feature for the purpose of investigating
the interplay between the mean flow and the turbulent drag of the bulk flow. By
means of a set of direct numerical simulations at increasing Reynolds number,
we show the dependence of the bulk turbulent drag on the amplitude of the mean
flow. Further, we present a detailed analysis of the scale-by-scale energy
balance, which describes how kinetic energy is redistributed among different
regions of the flow while being transported toward small dissipative scales.
Our results allow us to obtain an accurate prediction for the spatial energy
transport at large scales.
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| Turbulent Fluid Acceleration Generates Clusters of Gyrotactic
Microorganisms
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| Filippo De Lillo, Massimo Cencini, William M. Durham,
Michael Barry, Roman Stocker, Eric Climent and Guido Boffetta
|
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|
Physical Review Letters 112, 044502 (2014).
The motility of microorganisms is often biased by gradients in physical and
chemical properties of their environment, with myriad implications on their
ecology. Here we show that fluid acceleration reorients gyrotactic plankton,
triggering small-scale clustering. We experimentally demonstrate this
phenomenon by studying the distribution of the phytoplankton within a rotating
tank and find it to be in good agreement with a new, generalized model of
gyrotaxis. When this model is implemented in a direct numerical simulation of
turbulent flow, we find that fluid acceleration generates multifractal plankton
clustering, with faster and more stable cells producing stronger clustering. By
producing accumulations in high-vorticity regions, this process is
fundamentally different from clustering by gravitational acceleration,
expanding the range of mechanisms by which turbulent flows can impact the
spatial distribution of active suspensions.
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| Geotropic tracers in turbulent flows: a proxy for fluid acceleration
|
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| F. De Lillo, M. Cencini, G. Boffetta and F. Santamaria
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Journal of Turbulence 14.7, 24 (2013).
We investigate the statistics of orientation of small, neutrally buoyant,
spherical tracers whose centre of mass is displaced from the geometrical
centre. If appropriate-sized particles are considered, a linear relation can be
derived between the horizontal components of the orientation vector and the
same components of acceleration. Direct numerical simulations are carried out,
showing that such relation can be used to reconstruct the statistics of
acceleration fluctuations up to the order of the gravitational acceleration.
Based on such results, we suggest a novel method for the local experimental
measurement of accelerations in turbulent flows.
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| A new assessment of the second-order moment of Lagrangian velocity increments in turbulence
|
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| A.S. Lanotte, L. Biferale, G. Boffetta and F. Toschi
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Journal of Turbulence 14.7, 34 (2013).
The behaviour of the second-order Lagrangian structure functions on
state-of-the-art numerical data both in two and three dimensions is studied. On
the basis of a phenomenological connection between Eulerian space-fluctuations
and the Lagrangian time-fluctuations, it is possible to rephrase the Kolmogorov
4/5-law into a relation predicting the linear (in time) scaling for the
second-order Lagrangian structure function. When such a function is directly
observed on current experimental or numerical data, it does not clearly display
a scaling regime. A parameterisation of the Lagrangian structure functions
based on Batchelor model is introduced and tested on data for 3d turbulence,
and for 2d turbulence in the inverse cascade regime. Such parameterisation
supports the idea, previously suggested, that both Eulerian and Lagrangian data
are consistent with a linear scaling plus finite-Reynolds number effects
affecting the small- and large timescales. When large-time saturation effects
are properly accounted for, compensated plots show a detectable plateau already
at the available Reynolds number. Furthermore, this parameterisation allows us
to make quantitative predictions on the Reynolds number value for which
Lagrangian structure functions are expected to display a scaling region.
Finally, we show that this is also sufficient to predict the anomalous
dependency of the normalised root mean squared acceleration as a function of
the Reynolds number, without fitting parameters.
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| Turbulence drives microscale patches of motile phytoplankton
|
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| William M. Durham, Eric Climent, Michael Barry, Filippo De Lillo,
Guido Boffetta, Massimo Cencini, and Roman Stocker
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|
Nature Communications 4, 2148 (2013).
Patchiness plays a fundamental role in phytoplankton ecology by dictating the
rate at which individual cells encounter each other and their predators. The
distribution of motile phytoplankton species is often considerably more patchy
than that of non-motile species at submetre length scales, yet the mechanism
generating this patchiness has remained unknown. Here we show that strong
patchiness at small scales occurs when motile phytoplankton are exposed to
turbulent flow. We demonstrate experimentally that Heterosigma akashiwo forms
striking patches within individual vortices and prove with a mathematical model
that this patchiness results from the coupling between motility and shear. When
implemented within a direct numerical simulation of turbulence, the model
reveals that cell motility can prevail over turbulent dispersion to create
strong fractal patchiness, where local phytoplankton concentrations are
increased more than 10-fold. This ‘unmixing’ mechanism likely enhances
ecological interactions in the plankton and offers mechanistic insights into
how turbulence intensity impacts ecosystem productivity.
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| Stokes drift for inertial particles transported by water waves
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| F. Santamaria, G. Boffetta, M. Martins Afonso, A. Mazzino,
M. Onorato, and D. Pugliese
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Europhysics Letters 102, 14003 (2013).
We study the effect of surface gravity waves on the motion of inertial
particles in an incompressible fluid. We perform analytical calculations
based on perturbation expansions which allow us to predict the dynamics
of inertial particles in the deep-water regime. We find that the presence
of inertia leads to a non-negligible correction to the well-known
horizontal Stokes drift velocity.
Moreover, we find that the vertical sedimentation velocity is also affected
by a drift induced by waves. The latter result may have some relevant
consequences on the rate of sedimentation of particles of finite size.
We underline that the vertical drift would also be observed in the
(hypothetical) absence of the gravitational force. Kinematic numerical
simulations are performed and the results are found to be in excellent
agreement with the analytical predictions, even for values of the
parameters beyond the perturbative limit.
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| Anomalous diffusion in confined turbulent convection
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| G. Boffetta, F. De Lillo, and S. Musacchio
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Physics Review E 85, 066322 (2012).
Turbulent convection in quasi-one-dimensional geometry is studied by means of
high-resolution direct numerical simulations within the framework of
Rayleigh-Taylor turbulence. Geometrical confinement has dramatic effects on the
dynamics of the turbulent flow, inducing a transition from superdiffusive to
subdiffusive evolution of the mixing layer and arresting the growth of kinetic
energy. A nonlinear diffusion model is shown to reproduce accurately the above
phenomenology. The model is used to predict, without free parameters, the
spatiotemporal evolution of the heat flux profile and the dependence of the
Nusselt number on the Rayleigh number.
|
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| On Lagrangian single-particle statistics
|
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| G. Falkovich, H. Xu, A. Pumir, E. Bodenschatz, L. Biferale,
G. Boffetta, A.S. Lanotte and F. Toschi
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Physics of Fluids 24, 055102 (2012).
In turbulence, ideas of energy cascade and energy flux, substantiated by the
exact Kolmogorov relation, lead to the determination of scaling laws for the
velocity spatial correlation function. Here we ask whether similar ideas can be
applied to temporal correlations. We critically review the relevant theoretical
and experimental results concerning the velocity statistics of a single fluid
particle in the inertial range of statistically homogeneous, stationary and
isotropic turbulence. We stress that the widely used relations for the second
structure function, D2(t) ≡ ⟨[v(t) − v(0)]2⟩∝εt, relies on dimensional
arguments only: no relation of D2(t) to the energy cascade is known, neither in
two- nor in three-dimensional turbulence. State of the art experimental and
numerical results demonstrate that at high Reynolds numbers, the derivative
math has a finite non-zero slope starting from t ≈ 2τη. The analysis of the
acceleration spectrum ΦA(ω) indicates a possible small correction with respect
to the dimensional expectation ΦA(ω) ∼ ω0 but present data are unable to
discriminate between anomalous scaling and finite Reynolds effects in the
second order moment of velocity Lagrangian statistics.
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| Unraveling the influence of endothelial cell density on
VEGF-A signaling
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| L. Napione, S. Pavan, A. Veglio, A. Picco, G. Boffetta, A. Celani,
G. Seano, L. Primo, A. Gamba and F. Bussolino
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|
Blood 10.1182/blood-2011-11-390666 (2012).
Vascular endothelial growth factor-A (VEGF) is the master determinant for the
activation of the angiogenic program leading to the formation of new blood
vessels to sustain solid tumor growth and metastasis. VEGF specific binding to
VEGF receptor-2 (VEGFR-2) triggers different signaling pathways including
phospholipase Cγ (PLCγ) and Akt cascades, crucial for endothelial
proliferation, permeability and survival. By combining biological experiments,
theoretical insights and mathematical modeling, we found that: (i) cell density
influences VEGFR-2 protein level, as receptor number is 2-fold higher in
long-confluent than in sparse cells; (ii) cell density affects VEGFR-2
activation by reducing its affinity for VEGF in long-confluent cells; (iii)
despite reduced ligand-receptor affinity, high VEGF concentrations provide
long-confluent cells with a larger amount of active receptors; (iv) PLCγ and
Akt are not directly sensitive to cell density, but simply transduce downstream
the upstream difference in VEGFR-2 protein level and activation; (v) the
mathematical model correctly predicts the existence of at least one protein
tyrosine phosphatase directly targeting PLCγ and counteracting the
receptor-mediated signal. Our data-based mathematical model quantitatively
describes VEGF signaling in quiescent and angiogenic endothelium, and is
suitable to identify new molecular determinants and therapeutic targets.
|
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| Control of particle clustering in turbulence by polymer
additives
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| F. De Lillo, G. Boffetta, S. Musacchio
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|
Physical Review E 85, 036308 (2012).
We study the clustering properties of inertial particles in a turbulent
viscoelastic fluid. The investigation is carried out by means of direct
numerical simulations of turbulence in the Oldroyd-B model. The effects of
polymers on the small-scale properties of homogeneous turbulence are considered
in relation with their consequences on clustering of particles, both lighter
and heavier than the carrying fluid. We show that, depending on particle and
flow parameters, polymers can either increase or decrease clustering.
|
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| Two-Dimensional Turbulence
|
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| G. Boffetta, R.E. Ecke
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Annual Review of Fluid Mechanics 44, 427 (2012).
In physical systems, a reduction in dimensionality often leads to exciting new
phenomena. Here we discuss the novel effects arising from the consideration of
fluid turbulence confined to two spatial dimensions. The additional
conservation constraint on squared vorticity relative to three-dimensional (3D)
turbulence leads to the dual-cascade scenario of Kraichnan and Batchelor with
an inverse energy cascade to larger scales and a direct enstrophy cascade to
smaller scales. Specific theoretical predictions of spectra, structure
functions, probability distributions, and mechanisms are presented, and major
experimental and numerical comparisons are reviewed. The introduction of 3D
perturbations does not destroy the main features of the cascade picture,
implying that 2D turbulence phenomenology establishes the general picture of
turbulent fluid flows when one spatial direction is heavily constrained by
geometry or by applied body forces. Such flows are common in geophysical and
planetary contexts, are beautiful to observe, and reflect the impact of
dimensionality on fluid turbulence.
|
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| Bolgiano scale in confined Rayleigh--Taylor turbulence
|
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| G. Boffetta, F. De Lillo, A. Mazzino and S. Musacchio
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Journal of Fluid Mechanics 690, 426 (2012).
We investigate the statistical properties of Rayleigh-Taylor turbulence
in a three-dimensional convective cell of high aspect ratio,
in which one transverse side is much smaller that the others.
By means of high-resolution numerical simulation we study the
development of the turbulent mixing layer
and the scaling properties of the velocity and temperature fields.
We show that the system undergoes a transition from
three- to two-dimensional turbulent regime
when the width of the turbulent mixing layer becomes larger than
the scale of confinement.
In the late stage of the evolution the convective flow
is characterized by the coexistence of Kolmogorov-Obukhov and
Bolgiano-Obukhov scaling at small and large scales, respectively.
These regimes are separated by the Bolgiano scale,
which is determined by the scale of confinement of the flow.
Our results show that the emergence of the Bolgiano-Obukhov scaling in
Rayleigh-Taylor turbulence is connected
to the onset of an upscale energy transfer induced
by the geometrical constraint of the flow.
|
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| The ultimate state of thermal convection in Rayleigh-Taylor
turbulence
|
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| G. Boffetta, F. De Lillo, A. Mazzino and L. Vozella
|
|
|
Physica D 241, 137 (2012).
This paper discusses the so-called ultimate state of thermal convection, first
proposed by R.H. Kraichnan almost 50 years ago and recently observed in
numerical simulations of turbulent convection in the absence of boundaries. We
focus on numerical simulations of turbulence generated by the Rayleigh–Taylor
instability in a wide range of Rayleigh and Prandtl numbers. Our results point
out to the conclusion that RT turbulence provides a natural realization of the
ultimate state of thermal convection thus highlighting the relationship between
the absence of boundaries and the emergence of the ultimate state scaling for
global statistical quantities.
|
|
| A flux loop mechanism in two-dimensional stratified turbulence
|
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| G. Boffetta, F. De Lillo, A. Mazzino and S. Musacchio
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|
Europhysics Letters 95, 34001 (2011).
We discuss the phenomenology of energy transfer in
two-dimensional, weakly stably stratified turbulence.
Kinetic energy, mechanically injected at small scales, is transferred
by turbulence towards large scales.
This inverse cascade proceeds up to the Ozmidov scale, where buoyancy
forces becomes effective. Kinetic energy is converted into potential energy,
which is transferred back towards small scales via a turbulent cascade of
density fluctuations.
The resulting "flux loop" is a novel mechanism which produces a non-trivial
stationary state in two-dimensional turbulence in the absence of a large
scale dissipation.
|
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| Shell model for quasi-two-dimensional turbulence
|
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| G. Boffetta, F. De Lillo and S. Musacchio
|
|
|
Physical Review E 83, 066302 (2011).
We discuss the possibility to introduce geometrical constraints in shell models
of turbulence in order to mimic the turbulent dynamics that takes place in
fluid layers with large aspect ratio. By using a scale-dependent set of
coupling parameters, we are able to resolve both scales larger and smaller than
a geometrical dimension of the flow. The proposed model is able to resolve with
high accuracy the split energy cascade phenomenon recently observed in such
flows, and allows us to investigate in detail the scaling properties of
turbulent convection confined in narrow convective cells.
|
|
| Effects of polymer additives on Rayleigh-Taylor turbulence
|
|
| G. Boffetta, A. Mazzino and S. Musacchio
|
|
|
Physical Review E 83, 056318 (2011).
The role of polymer additives on the turbulent convective flow of a
Rayleigh-Taylor system is investigated by means of direct numerical simulations
of Oldroyd-B viscoelastic model. The dynamics of polymer elongations follows
adiabatically the self-similar evolution of the turbulent mixing layer and
shows the appearance of a strong feedback on the flow which originates a cutoff
for polymer elongations. The viscoelastic effects on the mixing properties of
the flow are twofold. Mixing is appreciably enhanced at large scales (the
mixing layer growth rate is larger than that of the purely Newtonian case) and
depleted at small scales (thermal plumes are more coherent with respect to the
Newtonian case). The observed speed up of the thermal plumes, together with an
increase of the correlations between temperature field and vertical velocity,
contributes to a significant enhancement of heat transport. Our findings are
consistent with a scenario of drag reduction induced by polymers. A weakly
nonlinear model proposed by Fermi for the growth of the mixing layer is
reported in the Appendix.
|
|
| Elastic waves and transition to elastic turbulence in a
two-dimensional viscoelastic Kolmogorov flow
|
|
| S. Berti and G. Boffetta
|
|
|
Physical Review E 82, 036314 (2010).
We investigate the dynamics of the two-dimensional periodic Kolmogorov flow of
a viscoelastic fluid, described by the Oldroyd-B model, by means of direct
numerical simulations. Above a critical Weissenberg number the flow displays a
transition from stationary to randomly fluctuating states, via periodic ones.
The increasing complexity of the flow in both time and space at progressively
higher values of elasticity accom- panies the establishment of mixing features.
The peculiar dynamical behavior observed in the simulations is found to be
related to the appearance of filamental propagating patterns, which develop
even in the limit of very small inertial nonlinearities, thanks to the feedback
of elastic forces on the flow.
|
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| Evidence for the double cascade scenario in two-dimensional
turbulence
|
|
| G. Boffetta and S. Musacchio
|
|
|
Physical Review E 82, 016307 (2010).
Statistical features of homogeneous, isotropic, two-dimensional turbulence is
discussed on the basis of a set of direct numerical simulations up to the
unprecedented resolution 32768^2. By forcing the system at intermediate scales,
narrow but clear inertial ranges develop both for the inverse and for direct
cascades where the two Kolmogorov laws for structure functions are
simultaneously observed. The inverse cascade spectrum is found to be consistent
with Kolmogorov-Kraichnan prediction and is robust with respect the presence of
an enstrophy flux. The direct cascade is found to be more sensible to finite
size effects: the exponent of the spectrum has a correction with respect
theoretical prediction which vanishes by increasing the resolution.
|
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| Polymer Heat Transport Enhancement in Thermal Convection:
the Case of Rayleigh-Taylor Turbulence
|
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| G. Boffetta, A. Mazzino, S. Musacchio and L. Vozella
|
|
|
Physical Review Letters 104, 184501 (2010).
We study the effects of polymer additives on turbulence generated by
the ubiquitous Rayleigh-Taylor instability. Numerical simulations of
complete viscoelastic models provide clear evidence that the heat
transport is enhanced up to 50% with respect to the Newtonian case.
This phenomenon is accompanied by a speed-up of the mixing layer growth.
We give a phenomenological interpretation of these results based
on small-scale turbulent reduction induced by polymers.
|
|
| Statistics of mixing in three-dimensional Rayleigh-Taylor
turbulence at low Atwood number and Prandtl number one
|
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| G. Boffetta, A. Mazzino, S. Musacchio and L. Vozella
|
|
|
Physics of Fluids 22, 035109 (2010).
Three-dimensional miscible Rayleigh--Taylor (RT) turbulence at
small Atwood number and at Prandtl number one is investigated by
means of high resolution direct numerical simulations of the
Boussinesq equations.
RT turbulence is a paradigmatic time-dependent turbulent system
in which the integral scale grows in time following the evolution
of the mixing region.
In order to fully characterize the statistical properties
of the flow, both temporal and spatial behavior of relevant
statistical indicators have been analyzed.
Scaling of both global quantities (e.g., Rayleigh, Nusselt and
Reynolds numbers) and scale dependent observables built in terms of
velocity and temperature fluctuations are considered.
We extend the mean-field analysis for velocity and temperature
fluctuations to take into account intermittency, both in time and
space domains.
We show that the resulting scaling exponents are compatible with
those of classical Navier--Stokes turbulence advecting a passive
scalar at comparable Reynolds number. Our
results support the scenario of universality of turbulence with respect
to both the injection mechanism and the geometry of the flow.
|
|
| Nonlinear diffusion model for Rayleigh-Taylor mixing
|
|
| G. Boffetta, F. De Lillo and S. Musacchio
|
|
|
Physical Review Letters 104, 034505 (2010).
The complex evolution of turbulent mixing in Rayleigh-Taylor
convection is studied in terms of eddy diffusiviy models for
the mean temperature profile.
It is found that a non-linear model, derived within the general
framework of Prandtl mixing theory, reproduces accurately the
evolution of turbulent profiles obtained from numerical simulations.
Our model allows to give very precise predictions for the turbulent heat
flux and for the Nusselt number in the ultimate state regime
of thermal convection.
|
|
| Rayleigh-Taylor instability in a viscoelastic binary fluid
|
|
| G. Boffetta, A. Mazzino, S. Musacchio and L. Vozella
|
|
|
Journal of Fluid Mechanics 643, 127 (2010).
The effects of polymer additives on Rayleigh-Taylor (RT)
instability of immiscible fluids is investigated using the
Oldroyd-B viscoelastic model. Analytic results obtained
exploiting the phase-field approach show that in polymer
solution the growth rate of the instability speeds up with
elasticity (but remains slower than in the pure solvent
case). Numerical simulations of the viscoelastic binary
fluid model confirm this picture.
|
|
| Modeling Kelvin Wave Cascades in Superfluid Helium
|
|
| G. Boffetta, A. Celani, D. Dezzani, J. Laurie and S. Nazarenko
|
|
|
Journal of Low Temperature Physics 156, 193 (2009).
We study two different types of simplified models for Kelvin wave
turbulence on quantized vortex lines in superfluids near zero temperature.
Our first model is obtained from a truncated expansion of the Local
Induction Approximation (Truncated-LIA) and it is shown to possess the
same scalings and the essential behaviour as the full Biot-Savart model,
being much simpler than the later and, therefore, more amenable to
theoretical and numerical investigations. The Truncated-LIA model supports
six-wave interactions and dual cascades, which are clearly demonstrated via
the direct numerical simulation of this model in the present paper.
In particular, our simulations confirm presence of the weak turbulence regime
and the theoretically predicted spectra for the direct energy cascade and
the inverse wave action cascade. The second type of model we study, the
Differential Approximation Model (DAM), takes a further drastic simplification
by assuming locality of interactions in k-space via using a differential
closure that preserves the main scalings of the Kelvin wave dynamics.
DAMs are even more amenable to study and they form a useful tool by
providing simple analytical solutions in the cases when extra physical effects
are present, e.g. forcing by reconnections, friction dissipation and phonon
radiation. We study these models numerically and test their theoretical
predictions, in particular the formation of the stationary spectra, and
closeness of numerics for the higher-order DAM to the analytical predictions
for the lower-order DAM.
|
|
| Kolmogorov scaling and intermittency in Rayleigh-Taylor turbulence
|
|
| G. Boffetta, A. Mazzino, S. Musacchio, and L. Vozella
|
|
|
Physical Review E 79, 065301 (2009).
Turbulence induced by Rayleigh-Taylor instability is a ubiquitous phenomenon
with applications ranging from atmospheric physics and geophysics to
supernova explosions and plasma confinement fusion. Despite its fundamental
character, a phenomenological theory has been proposed only recently and
several predictions are untested. In this Rapid Communication we confirm
spatiotemporal predictions of the theory by means of direct numerical
simulations at high resolution and we extend the phenomenology to take
into account intermittency effects. We show that scaling exponents are
indistinguishable from those of Navier-Stokes turbulence at comparable
Reynolds number, a result in support of the universality of turbulence
with respect to the forcing mechanism. We also show that the time
dependence of Rayleigh, Reynolds, and Nusselt numbers realizes the Kraichnan
scaling regime associated with the ultimate state of thermal convection.
|
|
| Peripheral mixing of passive scalar at small Reynolds number
|
|
| G. Boffetta, F. De Lillo and A. Mazzino
|
|
|
Journal of Fluid Mechanics 624, 151 (2009).
Mixing of a passive scalar in the peripheral region close to a wall is
investigated by means of accurate direct numerical simulations of both
a three-dimensional Couette channel flow at low Reynolds numbers and a
two-dimensional synthetic flow. In both cases, the resulting phenomenology
can be understood in terms of the theory recently developed by Lebedev and
Turitsyn (Phys. Rev. E, vol. 69, 2004, 036301). Our results
prove the robustness of the identified mechanisms responsible for the
persistency of scalar concentration close to the wall with important
consequences in completely different fields ranging from microfluidic
applications to environmental dispersion modelling.
|
|
| Twenty-five years of multifractals in fully developed turbulence:
a tribute to Giovanni Paladin
|
|
| G. Boffetta, A. Mazzino and A. Vulpiani
|
|
|
Journal of Physics A 41, 363001 (2008).
The paper On the multifractal nature of fully developed turbulence
and chaotic systems, by R. Benzi et al. published in this journal
in 1984 ( vol 17, page 3521) has been a starting point of many
investigations on the different faces of selfsimilarity and
intermittency in turbulent phenomena.
Since then, the multifractal model has become a useful tool for
the study of small scale turbulence,
in particular for detailed predictions of different
Eulerian and Lagrangian statistical properties.
In the occasion of the 50-th birthday of our unforgettable friend
and colleague Giovanni Paladin (1958-1996), we review here the basic concepts
and some applications of the multifractal model for turbulence.
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| Lagrangian statistics in two-dimensional free
turbulent convection
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| A. Bistagnino and G. Boffetta
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New Journal of Physics 10, 075018 (2008).
We discuss single-particle and two-particle statistics in two-dimensional
turbulent convection in the Bolgiano–Oboukhov regime by means
of high-resolution direct numerical simulations. Relative separation of two
particles is found to be described well by a generalization of the Richardson
diffusion model. Single-particle velocity structure functions are dominated by
large-scale eddies and therefore a careful analysis based on ‘exit-time’
statistics is necessary to identify turbulent contributions.
Because the velocity field is not intermittent, small-scale acceleration
statistics is found to be in good agreement with simple dimensional predictions.
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| Two-dimensional elastic turbulence
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| S. Berti, A. Bistagnino, G. Boffetta, A. Celani and S. Musacchio
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Physical Review E 77, 055306(R) (2008).
We report numerical evidence of elastic turbulence phenomenology in a
two-dimensional periodic Kolmogorov flow. By direct numerical simulations
of the Oldroyd-B viscoelastic model at very small Reynolds numbers, we
find that above the elastic instability threshold the flow develops an
elastic turbulent regime. We observe that both the turbulent drag and
the Lyapunov exponent increase with the Weissenberg number, indicating
the presence of a disordered, turbulentlike mixing flow. The energy
spectrum develops a power-law scaling range with an exponent close to
the experimental and theoretical expectations.
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| How winding is the coast of Britain?
Conformal invariance of rocky shorelines
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| G. Boffetta, A. Celani, D. Dezzani and A. Seminara
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Geophysical Research Letters 35, L03615 (2008).
We show that rocky shorelines with fractal dimension
4/3 are conformally invariant curves by measuring the
statistics of their winding angles from global high-resolution
data. Such coastlines are thus statistically equivalent to the
outer boundary of the random walk and of percolation clusters.
A simple model of coastal erosion gives an explanation for
these results. Conformal invariance allows also to predict
the highly intermittent spatial distribution of the flux of
pollutant diffusing ashore.
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| Nonlinear dynamics of the viscoelastic Kolmogorov flow
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| A. Bistagnino, G. Boffetta, A. Celani, A. Mazzino, A. Puliafito
and M. Vergassola
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Journal of Fluid Mechanics 590, 61 (2007).
The weakly nonlinear dynamics of large-scale perturbations in a viscoelastic
flow is investigated both analytically, via asymptotic methods, and
numerically. For sufficiently small elasticities, dynamics is ruled by a
Cahn-Hilliard equation with a quartic potential. Physically, this amounts to
saying that, for small elasticities, polymers do not alter the purely
hydrodynamical mechanisms responsible for the nonlinear dynamics in the
Newtonian case (i.e. without polymers). The approach to the steady state
is quantitatively similar to the Newtonian case as well, the dynamics being
ruled by the same kink-antikink interactions as in the Newtonian limit.
The above scenario does not extend to large elasticities. We found a
critical value above which polymers drastically affect the dynamics of
large-scale perturbations. In this latter case, a new dynamics not
observed in the Newtonian case emerges. The most evident fingerprint of
the new dynamics is the slowing down of the annihilation processes which lead
to the steady states via weaker kink-antikink interactions. In conclusion,
polymers strongly affect the large-scale dynamics. This takes place via a
reduction of drag forces we were able to quantify from the asymptotic
analysis. This suggests a possible relation of this phenomenon with the
dramatic drag-reduction effect taking place in the far turbulent regime.
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| Energy and enstrophy fluxes in the double cascade of 2d turbulence
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| G. Boffetta
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Journal of Fluid Mechanics 589, 253 (2007).
High resolution direct numerical simulations of two-dimensional
turbulence in stationary conditions are presented. The development
of an energy-enstrophy double cascade is studied and found to
be compatible with the classical Kraichnan theory in the limit
of extended inertial ranges.
The analysis of the joint distribution of energy and enstrophy fluxes
in physical space reveals a small value of cross correlation.
This result supports many experimental and numerical
studies where only one cascade is generated.
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| The Eulerian description of dilute collisionless suspension
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| G. Boffetta, A. Celani, F. De Lillo and S. Musacchio
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Europhysics Letters 78, 14001 (2007).
We analyze the statistical properties of an
Eulerian fluid model describing the evolution
of a suspension of inertial particles in an incompressible flow.
Regularity and compressibility of the velocity field for the inertial phase
are investigated in the limit of heavy particles
by means of numerical simulations in two- and three-dimensional flows.
We show that in the small Stokes number regime the Eulerian fluid model
is able to capture fine details of the clustering dynamics,
and exhibits good agreement with fully Lagrangian simulations of
inertial particle trajectories. The fluid description breaks down due
to collisions at Stokes numbers $\gtrsim 0.1$, the actual value depending
on the carrier flow characteristics.
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| Lagrangian velocity structure functions in Bolgiano turbulence
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| A. Bistagnino, G. Boffetta and A. Mazzino
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Physics of Fluids 19, 011703 (2007).
Single-particle Lagrangian velocity statistics in the
Bolgiano--Obukhov regime of
two-dimensional turbulent convection is investigated.
At variance with flows displaying the classical K41 phenomenology,
here, the leading contribution to the Lagrangian velocity fluctuations
is given by the largest eddies. This implies a linear behavior
in time for a typical velocity fluctuation in the time interval $t$.
The contribution to the Lagrangian velocity fluctuations
of local eddies (i.e. with a characteristic time of order $t$),
whose space/time scalings are ruled by the Bolgiano--Obukhov
theory, is thus not detectable
by standard Lagrangian statistical observables. To disentangle
contributions arising from the large eddies from those of local eddies,
a strategy based on exit-time statistics has successfully been exploited.
Lagrangian velocity increments in Bolgiano convection thus provide
a physically relevant example of
signal with \emph{more than smooth} fluctuations.
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| Inverse turbulent cascades and conformally invariant curves
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| D. Bernard, G. Boffetta, A. Celani and G. Falkovich
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Physical Review Letters 98, 024501 (2007).
We offer a new example of conformal invariance far from
equilibrium --- the inverse cascade of Surface Quasi-Geostrophic
(SQG) turbulence. We show that temperature isolines are
statistically equivalent to curves that can be mapped into a
one-dimensional Brownian walk (called Schramm-Loewner Evolution or
SLE$_{\kappa}$). The diffusivity is close to $\kappa=4$, that is
iso-temperature curves belong to the same universality class as
domain walls in the $O(2)$ spin model. Several statistics of
temperature clusters and isolines are measured and shown to be
consistent with the theoretical expectations for such a spin
system at criticality. We also show that the direct cascade in
two-dimensional Navier-Stokes turbulence is not conformal
invariant. The emerging picture is that conformal invariance may
be expected for inverse turbulent cascades of strongly
interacting systems.
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| Small scale statistics of viscoelastic turbulence
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| S. Berti, A. Bistagnino, G. Boffetta, A. Celani and S. Musacchio
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Europhysics Letters 76, 63 (2006).
The small scale statistics of homogeneous isotropic turbulence of
dilute polymer solutions is investigated by means of direct numerical
simulations of a simplified viscoelastic fluid model.
It is found that polymers only partially suppress the turbulent cascade
below the Lumley scale, leaving a remnant energy flux even for large
elasticity. As a consequence, fluid acceleration in viscoelastic flows
is reduced with respect to Newtonian turbulence, whereas its rescaled
probability density is left unchanged.
At large scales the velocity field is found to be unaffected by
the presence of polymers.
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| Lyapunov exponents of heavy particles in turbulence
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| J. Bec, L. Biferale, G. Boffetta, M. Cencini, S. Musacchio and F. Toschi
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I>Physics of Fluids 18, 091702 (2006).
Lyapunov exponents of heavy particles and tracers advected by homogeneous
and isotropic turbulent flows are investigated by means of direct numerical
simulations. For large values of the Stokes number, the main effect of
inertia is to reduce the chaoticity with respect to fluid tracers.
Conversely, for small inertia, a counterintuitive increase of the first
Lyapunov exponent is observed. The flow intermittency is found to induce a
Reynolds number dependency for the statistics of the finite-time Lyapunov
exponents of tracers. Such intermittency effects are found to persist at
increasing inertia.
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| Dynamics and statistics of heavy particles in turbulent
flows
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| M. Cencini, J. Bec, L. Biferale, G. Boffetta, A. Celani,
A. Lanotte, S. Musacchio and F. Toschi
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Journal of Turbulence 7, N36 (2006).
We present the results of direct numerical simulations (DNS) of turbulent
flows seeded with millions of passive inertial particles. The maximum
Reynolds number is Re_{\lambda}=200. We consider particles
much heavier than the carrier flow in the limit when the Stokes drag force
dominates their dynamical evolution.We discuss both the transient and the
stationary regimes. In the transient regime,we study the growth of
inhomogeneities in the particle spatial distribution driven by the
preferential concentration out of intense vortex filaments.
In the stationary regime, we study the acceleration fluctuations as a
function of the Stokes number in the range St=[0.16,3.3].
We also compare our results with those of pure fluid tracers (St = 0)
and we find a critical behavior of inertia for small Stokes values.
Starting from the pure monodisperse statistics we also characterize
polydisperse suspensions with a given mean Stokes, St.
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| Numerical studies of turbulent particle fluxes
into perfectly absorbing spherical surfaces
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| G. Boffetta, H.L. Pecseli and J. Trulsen
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Journal of Turbulence 7, N22 (2006).
With reference to studies of the influence of turbulence on
the feeding process of aquatic micro-organisms, we analyze particle
fluxes into absorbing surfaces in turbulent flows by numerical simulations.
The simultaneous trajectories of many point particles are followed
in time in a fully three-dimensional solution of the turbulent flow
described by the Navier Stokes equation.
Selecting one of these points to represent a predator, while the
others are considered as prey, we obtain estimates for the time
variation of the statistical average of particle fluxes into a
co-moving sphere of interception. The essential restriction in the model,
when applied to aquatic micro-organisms, is that self-induced motions
are ignored. Particles are assumed to be absorbed when crossing the surface.
In this sense, the problem can be analyzed as the one involving a
perfectly absorbing surface. The variation of the particle flux with
the radius in the absorbing sphere, as well as the variation with basic
flow parameters is well described by a simple scaling law, expressed in
terms of the radius of the sphere and the energy dissipated per mass unit.
The results also agree well with experimental results. In the present
study, we obtain a unique signal-to-noise ratio in the estimates.
The analysis is extended by inclusion of another dataset, with a somewhat
smaller Reynolds number. The scaling laws obtained by a simple dimensional
reasoning agree well for the two datasets. The numerical simulations refer
to two different Reynolds numbers, but the scaling laws verified for these
conditions can then be applied generally for other flows, provided the
basic assumptions are fulfilled: the turbulence has to be fully developed so
that a universal subrange exists, and the spatial scales defined by the
radii of the absorbing spherical surfaces have to be restricted to this
subrange.
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| Conformal invariance in two-dimensional turbulence
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| D. Bernard, G. Boffetta, A. Celani and G. Falkovich
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Nature Physics 2, 124 (2006).
Simplicity of fundamental physical laws manifests itself in fundamental
symmetries. While systems with an infinity of strongly interacting degrees
of freedom (in particle physics and critical phenomena) are hard to
describe, they often demonstrate symmetries, in particular scale invariance.
In two dimensions (2d) locality often promotes scale invariance to a
wider class of conformal transformations which allow for nonuniform rescaling.
Conformal invariance allows a thorough classification of universality
classes of critical phenomena in 2d.
Is there conformal invariance in 2d turbulence, a paradigmatic example
of strongly-interacting non-equilibrium system? Here, using numerical
experiment, we show that some features of 2d inverse turbulent cascade
display conformal invariance.
We observe that the statistics of vorticity clusters is remarkably close
to that of critical percolation, one of the simplest universality classes
of critical phenomena. These results represent a new step in the
unification of 2d physics within the framework of conformal symmetry.
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| Lagrangian statistics in fully developed turbulence
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| L. Biferale, G. Boffetta, A. Celani, A. Lanotte and F. Toschi
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Journal of Turbulence 7, N6 (2006).
The statistics of Lagrangian particles transported by a three dimensional
fully developed turbulent flow is investigated by means of high
resolution direct numerical simulations.
The analysis of single trajectories reveals the existence of
strong trapping events vortices at the Kolmogorov scale which contaminates
inertial range statistics up to $10 \tau_{\eta}$. For larger
time separations we find that Lagrangian structure functions
display intermittency in agreement with the prediction of the multifractal
model of turbulence.
The study of two-particle dispersion shows that the probability
density function of pair separation is very close to the
original prediction of Richardson of 1926.
Nevertheless, moments of relative dispersion are
strongly affected by finite Reynolds effects, thus limiting the
possibility to measure numerical prefactors, such as
the Richardson constant $g$. We show how, by
using an exit time statistics, it is possible to have a precise
estimation of $g$ which is consistent with recent laboratory measurements.
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| Multifractal clustering of passive tracers on a surface flow
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| G. Boffetta, J. Davoudi and F. De Lillo
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Europhysics Letters 74, 62 (2006).
We study the anomalous scaling of the mass density measure of Lagrangian
tracers in a compressible flow realized on the free surface on top of a
three dimensional flow.
The full two dimensional probability distribution of local stretching rates
is measured. The intermittency exponents which quantify the fluctuations of
the mass measure of tracers at small scales are calculated from the large
deviation form of stretching rate fluctuations.
The results indicate the existence of a critical exponent $n_c \simeq 0.86$
above which exponents saturate, in agreement with what has been predicted
by an analytically solvable model.
Direct evaluation of the multi-fractal dimensions by reconstructing the
coarse-grained particle density supports the results for low order moments.
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| Acceleration statistics of heavy particles in turbulence
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| J. Bec, L. Biferale, G. Boffetta, A. Celani, M. Cencini,
A. Lanotte, S. Musacchio and F. Toschi
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Journal of Fluid Mechanics 550, 349 (2006).
We present the results of direct numerical simulations of heavy
particle transport in homogeneous, isotropic, fully developed
turbulence, up to resolution $512^3$ ($R_\lambda\approx 185$).
Following the trajectories of up to 120 million particles with Stokes
numbers, $St$, in the range from $0.16$ to $3.5$ we are able to
characterize in full detail the statistics of particle
acceleration. We show that: ({\it i}\/) The root-mean-squared
acceleration $a_{\rm rms}$ sharply falls off from the fluid tracer
value already at quite small Stokes numbers; ({\it ii}\/) At a given
$St$ the normalised acceleration $a_{\rm rms}/(\epsilon^3/\nu)^{1/4}$
increases with $R_\lambda$ consistently with the trend observed for
fluid tracers; ({\it iii}\/) The tails of the probability density
function of the normalised acceleration $a/a_{\rm arms}$ decrease with
$St$. Two concurrent mechanisms lead to the above results: particle
clustering, very effective at small $St$, and filtering induced by the
particle response time, that takes over at larger $St$.
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| Turbulence and coarsening in active and passive binary
mixtures
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| S. Berti, G. Boffetta, M. Cencini and A. Vulpiani
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Physical Review Letters 95, 224501 (2005).
We address the problem of phase separation dynamics of two-dimensional
binary mixtures in the presence of external stirring. Both active and
passive mixtures are investigated. The phenomenon of coarsening arrest,
i.e. the appearance of a nontrivial stationary state with domains having a
finite length depending on the stirring intensity, is shown to be generic in
both chaotic and regular flows.
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| Lagrangian statistics of particle pairs in homogeneous
isotropic turbulence
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| L. Biferale, G. Boffetta, A. Celani, B.J. Devenish,
A. Lanotte and F. Toschi
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Physics of Fluids 17, 115101 (2005).
We present a detailed investigation of the particle pair separation
process in homogeneous isotropic turbulence. We use data from direct
numerical simulations up to $R_{\lambda} \sim 280$ following the
evolution of about two million passive tracers advected by the flow
over a time span of about three decades. We present data for both the
separation distance and the relative velocity statistics. Statistics
are measured along the particle pair trajectories both as a function
of time and as a function of their separation, i.e. at fixed
scales. We compare and contrast both sets of statistics in order to
gain an insight into the mechanisms governing the separation
process. We find very high levels of intermittency in the early
stages, that is, for travel times up to order ten Kolmogorov time
scales. The fixed scale statistics allow us to quantify anomalous
corrections to Richardson diffusion in the inertial range of scales
for those pairs that separate rapidly. It also allows a quantitative
analysis of intermittency corrections for the relative velocity
statistics.
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| Acceleration and vortex filaments in turbulence
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| F. Toschi,L. Biferale, G. Boffetta, A. Celani, B.J. Devenish and
A. Lanotte
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Journal of Turbulence 6, N15 (2005).
We report recent results from a high-resolution numerical study of
fluid particles transported by a fully developed turbulent flow.
Single-particle trajectories were followed for a time range spanning more
than three decades, from less than a tenth of the Kolmogorov timescale
up to one large-eddy turnover time. We present some results concerning
acceleration statistics and the statistics of trapping by vortex
filaments.
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| Multi-particle dispersion in fully developed turbulence
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| L. Biferale, G. Boffetta, A. Celani, B.J. Devenish,
A. Lanotte and F. Toschi
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Physics of Fluids 17, 111701 (2005).
The statistical geometry of dispersing Lagrangian clusters of
four particles (tetrahedra) is studied by means of high-resolution
direct numerical
simulations of three-dimensional homogeneous isotropic turbulence.
We give the first evidence of a self-similar regime of shape dynamics
characterized by almost two-dimensional, strongly elongated geometries.
The analysis of four-point velocity-difference statistics and
orientation shows that inertial-range eddies typically generate a
straining field
with a strong extensional component aligned with the elongation direction
and weak extensional/compressional components in the orthogonal plane.
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| Effects of friction on 2D turbulence:
An experimental study of the direct cascade
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| G. Boffetta, A. Cenedese, S. Espa and S. Musacchio
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Europhysics Letters 71, 590 (2005).
We study the direct enstrophy cascade
in a two-dimensional flow generated
in an electromagnetically driven thin layer of fluid.
Due to the presence of bottom friction,
the energy spectrum deviates
from the classical Kraichnan prediction $k^{-3}$.
We find that the correction to the spectral slope depends
on the thickness on the layer, in agreement with a theoretical
prediction based on the analogy with passive scalar statistics.
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| Drag reduction in the turbulent Kolmogorov flow
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| G. Boffetta, A. Celani and A. Mazzino
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Physical Review E 71, 036307 (2005).
We investigate the phenomenon of drag reduction in a viscoelastic
fluid model of dilute polymer solutions. By means of direct numerical
simulations of the three-dimensional turbulent Kolmogorov flow we show
that drag reduction takes place above a critical Reynolds number
$Re_c$. An explicit expression for the dependence of $Re_c$ on polymer
elasticity and diffusivity is derived. The values of the drag coefficient
obtained for different fluid parameters collapse onto a universal curve
when plotted as a function of the rescaled Reynolds number $Re/Re_c$.
The analysis of the momentum budget allows to gain some insight on the
physics of drag reduction, and suggests the existence of a $Re$-independent
value of the drag coefficient - lower than the Newtonian one -
for large Reynolds numbers.
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| Particle trapping in three-dimensional fully developed turbulence
F
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| L. Biferale, G. Boffetta, A. Celani, A. Lanotte and F. Toschi
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Physics of Fluids 17, 021701 (2005).
The statistical properties of fluid particles transported by a three
dimensional fully developed turbulent flow are investigated by means
of high resolution direct numerical simulations. Single trajectory
statistics are investigated in a time range spanning more than three
decades, from less than a tenth of the Kolmogorov timescale,
$\tau_{\eta}$, up to one large-eddy turnover time.
Our analysis reveal the existence of relatively rare
trapping events in vortex filaments which give rise to enhanced
intermittency on Lagrangian observables up to $10
\tau_{\eta}$. Lagrangian velocity structure function attain scaling
properties in agreement with the multifractal prediction only for
time lags larger than those affected by trapping.
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| The viscoelastic Kolmogorov flow: eddy-viscosity and
linear stability
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| G. Boffetta, A. Celani, A. Mazzino, A. Puliafito and M. Vergassola
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Journal of Fluid Mechanics 523, 161 (2005).
The stability properties of the laminar Kolmogorov flow
of a viscoelastic Oldroyd-B fluid are investigated
both analytically and numerically.
Linear stability with respect to large-scale perturbations
is studied by means of multiple-scale analysis.
This technique yields an effective diffusion equation for the large-scale
perturbation where the effective (eddy) viscosity
can be computed analytically.
Stability analysis is thus reduced to study the positive definiteness
of the eddy-viscosity tensor
as a function of the Reynolds and Deborah numbers.
Two main results emerge
from our analysis: {\it (i)\/}
at small fluid elasticity the flow is more stable than
in the Newtonian case; {\it (ii)\/} at large elasticity the flow
is prone to purely elastic instabilities (i.e. occurring
at zero Reynolds number).
The hypothesis of scale separation is very well verified up to
moderate elasticity, as checked by numerical integration
of the exact linearized equations by means of the Arnoldi method.
Finally, it is shown that the addition of a stress diffusivity
counteracts the effect of elasticity, in agreement with simple
physical arguments.
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| Lagrangian tracers on a surface flow:
the role of time correlations
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| G. Boffetta, J. Davoudi, B. Eckhardt and J. Schumacher
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Physical Review Letters 93, 134501 (2004).
Finite time correlations of the velocity in a surface flow
are found to be important for the formation of clusters of Lagrangian tracers.
The degree of clustering characterized
by the Lyapunov spectrum of the flow is numerically shown to be in qualitative
agreement with the predictions for the white-in-time compressible
Kraichnan flow, but to deviate quantitatively.
For intermediate values of compressibility the clustering is
surprisingly weakened by time correlations.
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| Multifractal statistics of Lagrangian velocity and
acceleration in turbulence
|
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| L. Biferale, G. Boffetta, A. Celani, B.J. Devenish, A. Lanotte
and F. Toschi
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Physical Review Letters 93, 064502 (2004).
The statistical properties of velocity and acceleration fields along
the trajectories of fluid particles transported by a fully developed
turbulent flow are investigated by means of high resolution direct
numerical simulations. We present results for Lagrangian velocity structure
functions, the acceleration probability density function and the
acceleration variance conditioned on the instantaneous velocity. These
are compared with predictions of the multifractal formalism and its
merits and limitations are discussed.
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| Large scale inhomogeneity of inertial particles in turbulent
flow
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| G. Boffetta, F. De Lillo and A. Gamba
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Physics of Fluids 16, L20 (2004).
Preferential concentration of inertial particles in
turbulent flow is studied by high resolution direct numerical simulations
of two-dimensional turbulence.
The formation of network-like regions of high particle density,
characterized by a length scale which depends on the Stokes number
of inertial particles, is observed.
At smaller scales, the size of empty regions appears to be distributed
according to a scaling law.
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| Two-dimensional turbulence of dilute polymer solutions
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| G. Boffetta, A. Celani and S. Musacchio
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Physical Review Letters 91, 034501 (2003).
We investigate theoretically and numerically the effect of polymer
additives on two-dimensional turbulence by means of a viscoelastic model.
We provide compelling evidence that at vanishingly small concentrations,
such that the polymers are passively transported, the probability
distribution of polymer elongation has a power law tail: its slope is
related to the statistics of finite-time Lyapunov exponents of the flow,
in quantitative agreement with theoretical predictions.
We show that at finite concentrations
and sufficiently large elasticity the polymers react on the flow
with manifold consequences: velocity fluctuations
are drastically depleted, as observed in soap film experiments;
the velocity statistics becomes strongly intermittent; the distribution
of finite-time Lyapunov exponents shifts to lower values,
signalling the reduction of Lagrangian chaos.
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| Relaxation of finite perturbations:
Beyond the Fluctuation-Response relation
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| G. Boffetta, G. Lacorata, S. Musacchio and A. Vulpiani
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Chaos 13, 806 (2003).
We study the response of dynamical systems to finite amplitude perturbation.
A generalized Fluctuation-Response relation is derived,
which links the average relaxation toward equilibrium
to the invariant measure of the system and points out the relevance
of the amplitude of the initial perturbation.
Numerical computations on systems with many characteristic times
show the relevance of the above relation in realistic cases.
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| The decay of homogeneous anisotropic turbulence
|
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| L. Biferale, G. Boffetta, A. Celani, A. Lanotte, F. Toschi and
M. Vergassola
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Physics of Fluids 15, 2105 (2003).
We present the results of a numerical investigation of
three-dimensional decaying turbulence with statistically homogeneous
and anisotropic initial conditions. We show that at large times, in
the inertial range of scales: (i) isotropic velocity fluctuations
decay self-similarly at an algebraic rate which can be obtained
by dimensional arguments; (ii) the ratio of anisotropic to isotropic
fluctuations of a given intensity falls off in time as a power law,
with an exponent approximately independent of the strength of the
fluctuation; (iii) the decay of anisotropic fluctuations is not
self-similar, their statistics becoming more and more intermittent as
time elapses. We also investigate the early stages of the decay.
The different short-time behavior observed in two experiments differing
by the phase organization of their initial conditions gives a new hunch on
the degree of universality of small-scale turbulence statistics, i.e. its
independence of the conditions at large scales.
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| Diffusive transport and self-consistent
dynamics in coupled maps
|
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| G. Boffetta, D. del-Castillo-Negrete, C. Lopez, G. Pucacco and A. Vulpiani
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Physical Review E 67 026224 (2003).
The study of diffusion in Hamiltonian systems has been a problem of
interest for a number of years.
In this paper we explore the influence of self-consistency on the
diffusion properties of systems described by coupled symplectic maps.
Self-consistency, i.e. the back-influence of the
transported quantity on the velocity field of the driving flow, despite of
its critical importance, is usually overlooked in the description of
realistic systems, for example in plasma physics.
We propose a class of self-consistent models consisting of an
ensemble of maps globally coupled through a mean field.
Depending on the kind of coupling, two different general types of
self-consistent maps are considered: maps coupled to the field only through
the phase, and fully coupled maps, i.e. through the phase and the amplitude
of the external field. The analogies and differences of the diffusion
properties of these two kinds of maps are discussed in detail.
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| Lagrangian Statistics and Temporal Intermittency in a
Shell Model of Turbulence
|
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| G. Boffetta, F. De Lillo and S. Musacchio
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Physical Review E 66, 066307 (2002).
We study the statistics of single particle Lagrangian velocity in
a shell model of turbulence.
We show that the small scale velocity fluctuations are
intermittent, with scaling exponents connected to the
Eulerian structure function scaling exponents.
The observed reduced scaling range is interpreted as a
manifestation of the intermediate dissipative range, as it
disappears in a Gaussian model of turbulence.
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| Structure functions and energy dissipation
dependence on Reynolds number
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| G. Boffetta and G.P. Romano
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Physics of Fluids 14, 3453 (2002).
The dependence of the statistics of energy dissipation on the Reynolds
number is investigated in an experimental jet flow.
In a range of about one decade of $Re_{\lambda}$ (from about 200 to 2000)
the adimensional mean energy dissipation is found to be independent on
$Re_{\lambda}$, while the higher moments of dissipation show a power-law
dependence. The scaling exponents are found to be consistent with a simple
prediction based on the multifractal model for inertial range structure
functions.
This is an experimental confirmation of the connection between inertial
range quantities and dissipation statistics predicted by the
multifractal approach.
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| Inverse cascade in Charney-Hasegawa-Mima turbulence
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| G. Boffetta, F. De Lillo and S. Musacchio
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Europhysics Letters 59, 687 (2002).
The inverse energy cascade in Charney-Hasegawa-Mima turbulence
is investigated. Kolmogorov law for the third order
velocity structure function is derived and shown to be independent on the
parameter $\lambda$, at variance with the energy spectrum, as shown
by high resolution direct numerical simulations.
In the asymptotic limit of strong rotation, $\lambda \to \infty$,
the Kolmogorov constant is found to be $C_{\lambda}\simeq 11$
while coherent vortices are observed to form at a dynamical scale
which slowly grows with time.
These vortices form an almost quenched pattern and induce strong
deviation form Gaussianity in the velocity field.
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| Intermittency in two-dimensional Ekman-Navier-Stokes
turbulence
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| G. Boffetta, A. Celani, S. Musacchio and M. Vergassola
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Physical Review E 66, 026304 (2002).
We study the statistics of the vorticity field in two-dimensional
Navier-Stokes turbulence with linear Ekman friction.
We show that the small-scale vorticity fluctuations are intermittent,
as conjectured by
D. Bernard, [{\it Europhys. Lett.} {\bf 50} (2000) 333] and
Nam {\it et al.\/} [{\it Phys. Rev. Lett.\/} {\bf 84} (2000) 5134].
The small-scale statistics of vorticity
fluctuations coincides with the one of a passive scalar with finite lifetime
transported by the velocity field itself.
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| Statistics of two-particle dispersion in two-dimensional
turbulence
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| G. Boffetta and I.M. Sokolov
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Physics of Fluids 14, 3224 (2002).
We investigate Lagrangian relative dispersion in direct numerical
simulation of two-dimensional inverse cascade turbulence.
The analysis is performed by using
both standard fixed time statistics and an exit time approach.
The latter allows a more precise determination of the Richardson
constant which is found to be $g \simeq 4$ with a possible weak
finite-size dependence.
Our results show only small deviations with respect to the original
Richardson's description in terms of diffusion equation.
These deviations are associated with the long-range correlated
nature of the particles' relative motion.
The correlation, or persistence, parameter is measured by means of
a Lagrangian ``turning point'' statistics.
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| Closure of two dimensional turbulence: the role of
pressure gradients
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| G. Boffetta, M. Cencini and J. Davoudi
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Physical Review E 66, 017301 (2002).
Inverse energy cascade regime of two dimensional turbulence is
investigated by means of high resolution numerical simulations.
Numerical computations of conditional averages of transverse pressure
gradient increments are found to be compatible with a recently proposed
self-consistent Gaussian model.
An analogous low order closure model for the longitudinal pressure gradient
is proposed and its validity is numerically examined. In this
case numerical evidence for the presence
of higher order terms in the closure is found.
The fundamental role of conditional statistics between longitudinal
and transverse components is highlighted.
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| Relative dispersion in fully developed turbulence:
The Richardson's Law and Intermittency Corrections
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| G. Boffetta and I.M. Sokolov
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Physical Review Letters 88, 094501 (2002).
Relative dispersion in fully developed turbulence is investigated
by means of direct numerical simulations. Lagrangian statistics
is found to be compatible with Richardson description although
small systematic deviations are found. The value of the Richardson
constant is estimated as $C_2 \simeq 0.55$, in a close agreement with
recent experimental findings [S. Ott and J. Mann J. Fluid Mech. {\bf 422},
207 (2000)]. By means of exit-time statistics it is shown that the deviations
from Richardson's law are a consequence of Eulerian intermittency.
The measured Lagrangian scaling exponents require a set of
Eulerian structure function exponents $\zeta _{p}$ which are remarkably
close to standard ones known for fully developed turbulence.
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| Predictability: a way to characterize Complexity
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| G. Boffetta, M. Cencini, M. Falcioni and A. Vulpiani
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Physics Reports 356, 367 (2002).
Different aspects of the predictability problem in dynamical systems
are reviewed. The deep relation among Lyapunov exponents,
Kolmogorov-Sinai entropy, Shannon entropy and algorithmic complexity
is discussed. In particular, we emphasize how a characterization of
the unpredictability of a system gives a measure of its complexity.
Adopting this point of view, we review some developments in
the characterization of the predictability of systems showing different
kind of complexity: from low-dimensional systems to high-dimensional
ones with spatio-temporal chaos and to fully developed turbulence.
A special attention is devoted to finite-time and finite-resolution effects
on predictability, which can be accounted with suitable generalization
of the standard indicators. The problems involved in systems
with intrinsic randomness is discussed, with emphasis on the important
problems of distinguishing chaos from noise and of modeling
the system. The characterization of irregular behavior in
systems with discrete phase space is also considered.
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| Detecting barriers to transport: A review of different
techniques
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| G. Boffetta, G. Lacorata, G. Redaelli and A. Vulpiani
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Physica D 159, 58 (2001).
We review and discuss some different techniques for
describing local dispersion properties in fluids.
A recent Lagrangian diagnostics, based on the Finite Scale
Lyapunov Exponent (FSLE), is presented and compared to
the Finite Time Lyapunov Exponent (FTLE), to the
Okubo-Weiss (OW) and Hua-Klein (HK) criteria.
We show that the OW and HK are a limiting case of the
FTLE, and that the FSLE is the most efficient method for
detecting the presence of cross-stream barriers.
We illustrate our findings by considering two examples of
geophysical interest: a kinematic model of meandering jet, and
Lagrangian tracers advected by stratospheric circulation.
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| Predictability of the inverse energy cascade in 2D
turbulence
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| G. Boffetta and S. Musacchio
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|
Physics of Fluids 13, 1060 (2001).
The predictability problem in the inverse energy cascade of
two-dimensional turbulence is addressed by means of high
resolution direct numerical simulations.
The growth rate as a function of the error level is determined
by means of a finite size extension of the Lyapunov exponent.
For errors within the inertial range, the
linear growth of the error energy, predicted by
dimensional argument, is verified with great accuracy.
Our numerical findings quantitatively confirms the results
of the classical TFM closure approximation.
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| Long-time behavior of MHD shell models
|
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| P. Frick, G. Boffetta, P. Giuliani, S. Lozhkin and D. Sokoloff
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Europhysics Letters 52, 539 (2000).
The long time behavior of velocity-magnetic field alignment
is numerically investigated in the framework of MHD shell model.
In the stationary forced case, the correlation parameter $C$
displays a nontrivial behavior with long periods of high
variability which alternates with periods of almost constant $C$.
The temporal statistics of correlation is shown
to be non Poissonian, and the pdf of constant sign periods displays
clear power law tails.
The possible relevance of the model for geomagnetic dynamo problem
is discussed.
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| Waiting time statistics in solar flares
|
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| P. Giuliani, V. Carbone, P. Veltri, G. Boffetta and A. Vulpiani
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Physica A 280, 75 (2000).
Solar flares activity is investigated by looking at the statistics of the
waiting times between hard X-ray bursts. It is found that the distribution
has a power-law tail which indicates the existence of
nontrivial dynamics with long-range correlations.
A shell model for MHD turbulence is capable
to reproduce the power-law distribution through an identification
of the intermittent bursts of
dissipation with solar flares.
This result suggests that the nonlinear dynamics could play a role
more relevant than the particular topology associated with the field
configuration. Comparisons
with results from models based on self-organized criticality are made.
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| Pair dispersion in turbulence
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| G. Boffetta and A. Celani
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Physica A 280, 1 (2000).
We study the statistics of pair dispersion in two-dimensional turbulence.
Direct numerical simulations show that
the pdf of pair separations is
in agreement with the Richardson prediction.
The pdf of doubling times follows dimensional scaling
and shows a long tail which is the signature of close approaches
between particles initially seeded with a large separation.
This phenomenon is related to the formation of fronts in
passive scalar advection.
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| Chaotic advection and relative dispersion in an experimental
convective flow
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| G. Boffetta, M. Cencini, S. Espa and G. Querzoli
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Physics of Fluids 12, 3160 (2000).
Lagrangian motion in a quasi-two dimensional, time dependent,
convective flow is studied at different Rayleigh numbers.
Particle Tracking Velocimetry technique is
used to reconstruct Lagrangian trajectories of passive tracers.
Dispersion properties are investigated by means of the recently
introduced finite size Lyapunov exponent analysis.
Lagrangian motion is found to be chaotic with a Lyapunov exponent
which depends on the Rayleigh number as ${\cal R}a^{1/2}$.
The power law scaling is explained in terms of a dimensional
analysis on the equation of motion.
A comparative study shows that the fixed scale method makes
more physical sense than the traditional way of looking at the
relative dispersion at fixed times.
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| Energy dissipation statistics in a shell model of
turbulence
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| G. Boffetta, A. Celani and D. Roagna
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Physical Review E 61, 3234 (2000).
The Reynolds number dependence of the statistics of energy
dissipation is investigated in a shell model of fully developed
turbulence.
The results are in agreement with a model which
accounts for fluctuations of the dissipative scale with
the intensity of energy dissipation.
It is shown that the assumption of a fixed dissipative scale
leads to a different scaling with Reynolds which is not compatible
with numerical results.
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| The predictability problem in systems with an uncertainty in the
evolution law
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| G. Boffetta, A. Celani, M. Cencini, G. Lacorata and A. Vulpiani
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Journal of Physics A 33, 1313 (2000).
The problem of unpredictability in a physical system due to the
incomplete knowledge of the evolution laws is addressed. Major
interest is devoted to the analysis of error amplification in chaotic
systems with many characteristic times and scales when the fastest
scales are not resolved.
The parameterization of the unresolved scales introduces a non
infinitesimal uncertainty (with respect the true evolution laws) which
affects the forecasting ability on the large resolved scales. The
evolution of non infinitesimal errors from the unresolved scales up to
the large scales is analyzed by means of the finite size Lyapunov
exponent. It is shown that proper parameterization of the unresolved
scales allows to recover the maximal predictability of the system.
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| Large scale instabilities in two-dimensional
magnetohydrodynamics
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| G. Boffetta, A. Celani and R. Prandi
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Physical Review E 61, 4329 (2000).
The stability of a sheared magnetic field is analyzed in two-dimensional
magnetohydrodynamics with resistive and viscous dissipation.
Using a multiple-scale analysis, it is shown that
at large enough Reynolds numbers the basic state describing a motionless
fluid and a layered magnetic field, becomes unstable
with respect to large scale perturbations.
The exact expressions for eddy-viscosity and eddy-resistivity are derived
in the nearby of the critical point where
the instability sets in.
In this marginally unstable case the nonlinear phase
of perturbation growth obeys to a Cahn-Hilliard-like dynamics
characterized by coalescence of magnetic islands leading to a final
new equilibrium state.
High resolution numerical simulations confirm quantitatively the
predictions of multiscale analysis.
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| Non asymptotic properties of transport and mixing
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| G. Boffetta, A. Celani, M. Cencini, G. Lacorata and A. Vulpiani
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Chaos 10, 50 (2000).
We study relative dispersion of passive scalar in non-ideal cases,
i.e. in situations in which asymptotic techniques cannot
be applied; tipically when the characteristic length scale of the Eulerian
velocity field is not much smaller than the domain size.
Of course, in such a situation usual asymptotic quantities (the
diffusion coefficients) do not give relevant informations about the
transport mechanisms. On the other hand, we shall show that the Finite
Size Lyapunov Exponent, originally introduced for the predictability
problem, appears to be a rather powerful approach to the non-asymptotic
transport properties. This technique is applied in a series of
numerical experiments in simple flows with chaotic behaviors, in
experimental data analysis of drifter and to study relative dispersion
in fully developed turbulence.
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| Inverse energy cascade in two-dimensional turbulence: Deviations fromGaussian behavior
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| G. Boffetta, A. Celani and M. Vergassola
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Physical Review E 61, R29 (2000).
High resolution numerical simulations of stationary inverse energy
cascade in two-dimensional turbulence are presented. Deviations from
Gaussianity of velocity differences statistics are quantitatively
investigated. The level of statistical convergence is pushed enough to
permit reliable measurement of the asymmetries in the probability
distribution functions of longitudinal increments and odd-order
moments, which bring the signature of the inverse energy flux. No
measurable intermittency corrections could be found in their scaling
laws. The seventh order skewness increases by almost two
orders of magnitude with respect to the third, thus becoming of order
unity.
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| Experimental evidence of chaotic advection in a convective
flow
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| G. Boffetta, M. Cencini, S. Espa and G. Querzoli
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|
Europhysics Letters 48, 629 (1999).
Lagrangian chaos is experimentally investigated in a convective flow
by means of Particle Tracking Velocimetry. The Finite Size
Lyapunov Exponent analysis is applied to quantify dispersion
properties at different scales.
In the range of parameters of the experiment, Lagrangian motion
is found to be chaotic.
Moreover, the Lyapunov depends on the Rayleigh number as Ra**0.5.
A simple dimensional argument for explaining the observed power
law scaling is proposed.
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| Power laws in solar flares: self-organized criticality or
turbulence?
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| G. Boffetta, V. Carbone, P. Giuliani, P. Veltri and A. Vulpiani
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|
Physical Review Letters 83, 4662 (1999).
We study the time evolution of Solar Flares activity by looking
at the statistics of quiescent times $\tau_{L}$ between
successive bursts.
The analysis of 20 years of data reveals a power law distribution
with exponent $\alpha \simeq 2.4$
which is an indication of complex dynamics with long correlation times.
The observed scaling behavior is in contradiction with the
Self-Organized Criticality models of Solar Flares which predict
Poisson-like statistics.
Chaotic models, including the destabilization of the laminar phases
and subsequent restabilization due to nonlinear dynamics, are able to
reproduce the power law for the quiescent times. In the case of
the more realistic Shell Model of MHD turbulence we are able to reproduce
all the observed distributions.
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| Pair dispersion in synthetic fully developed turbulence
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| G. Boffetta, A. Celani, A. Crisanti and A. Vulpiani
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|
Physical Review E 60, 6734 (1999).
The Lagrangian statistics of relative dispersion in fully developed
turbulence is numerically investigated.
A scaling range spanning many decades is achieved by generating a
two-dimensional velocity field by means of a stochastic process with
prescribed statistics and of a dynamical model (Shell Model) with
fluctuating characteristic times.
When the velocity field obeys Kolmogorov similarity
the Lagrangian statistics is self similar and
agrees with Richardson's predictions.
For intermittent velocity fields the scaling laws for the
Lagrangian statistics are found to depend
on the Eulerian intermittency in agreement with the multifractal
description.
As a consequence of the Kolmogorov law
the Richardson law for the variance of pair separation is,
however, not affected by intermittency corrections.
Moreover, Lagrangian exponents do not depend on the particular
Eulerian dynamics.
A new method of data analysis, based on fixed scale statistics
rather than usual
fixed time statistics, is shown to give much wider scaling
range and should be preferred for the analysis of experimental data.
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| Intermittency of Two-Dimensional Decaying Electron
Magnetohydrodynamic Turbulence
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| G. Boffetta, A. Celani, A. Crisanti and R. Prandi
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|
Physical Review E 59, 3726 (1999).
The intermittent nature of energy dissipation in two-dimensional
electron-MHD turbulence is investigated by means of high resolution direct
numerical simulations. It is found that, when the main contribution to
the energy is given by the magnetic field,
dissipation is mostly concentrated on one-dimensional
filaments. As a consequence, the multifractal spectrum has a simple
form which can be approximately described in terms of a bifractal model.
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| Relative dispersion in fully developed turbulence:
Lagrangian statistics in synthetic flows
|
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| G. Boffetta, A. Celani, A. Crisanti and A. Vulpiani
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|
Europhysics Letters 46, 177 (1999).
The effect of Eulerian intermittency on the Lagrangian statistics of
relative dispersion in fully developed turbulence is investigated.
A scaling range spanning many decades is achieved by generating a
multi-affine synthetic velocity field with prescribed intermittency
features.
The scaling laws for the Lagrangian statistics are found to depend
on intermittency in agreement with a multifractal
description. As a consequence of the Kolmogorov law,
the Richardson law for the variance of pair separation is not affected
by intermittency corrections.
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| Multi-time, multi-scale correlation functions in turbulence
and in turbulent models
|
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| L. Biferale, G. Boffetta, A. Celani and F. Toschi
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Physica D 127, 187 (1999).
A multifractal-like representation for multi-time multi-scale velocity
correlation in turbulence and dynamical turbulent models
is proposed.
The importance of subleading contributions to time correlations
is highlighted. The fulfillment of the dynamical constraints due
to the equations of motion is thoroughly discussed.
The predictions stemming from this representation are
tested within the framework of shell models for turbulence.
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| Predictability in chaotic systems and turbulence
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| G. Boffetta and A. Celani
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Journal de Physique IV France 8, Pr6 (1998).
A method for characterizing the predictability of complex
chaotic systems based on a generalization of the Lyapunov
exponent is introduced. The method is illustrated on a
toy system with two time scales and on a model of fully
developed turbulence where universal features are found.
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| Mimicking a turbulent signal: sequential multiaffine
processes
|
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| L. Biferale, G. Boffetta, A. Celani, A. Crisanti and A. Vulpiani
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|
Physical Review E 57, R6261 (1998).
An efficient method for the construction of a multiaffine
process, with prescribed scaling exponents, is presented.
At variance with the previous proposals, this method is sequential
and therefore it is the natural candidate in numerical computations
involving synthetic turbulence.
The application to the realization of a realistic turbulent-like
signal is discussed in detail.
The method represents a first step towards the realization of a
realistic spatio-temporal turbulent field.
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| Kolmogorov's law for two-dimensional electron-magnetohydrodynamic
turbulence
|
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| A. Celani, R. Prandi and G. Boffetta
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|
Europhysics Letters 41, 13-18 (1998).
The analogue of the Kolmogorov's four-fifths law
is derived
for two-dimensional, homogeneous, isotropic EMHD turbulence
in the energy cascade inertial range.
Direct numerical simulations for the freely decaying case
show that this relation holds true for different values
of the adimensional electron inertial length scale, $d_e$.
The energy spectrum is found to be close to the expected
Kolmogorov spectrum.
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| Slow and fast dynamics in coupled systems: A time series analysis
view
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| G. Boffetta, A. Crisanti, F. Paparella, A. Provenzale and A. Vulpiani
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Physica D 116, 301-312 (1998).
We study the dynamics of systems with different time
scales, when access only to the slow variables is allowed. We use
the concept of Finite Size Lyapunov Exponent (FSLE) and consider
both the case when the equations of motion for the slow components
are known, and the situation when a scalar time series of
one of the slow variables has been measured. A discussion on the
effects of parameterizing the fast dynamics is given.
We show that, although the computation of the largest
Lyapunov exponent can be practically infeasible in complex
dynamical systems, the computation of the FSLE allows to extract
information on the characteristic time and on the predictability of the
large-scale, slow-time dynamics even with moderate statistics and
unresolved small scales.
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| An extension of the Lyapunov analysis for the predictability
problem
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| G. Boffetta, P. Giuliani, G. Paladin and A. Vulpiani
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|
Journal of the Atmospheric Sciences 55, 3409-3416 (1998).
The predictability problem for systems with different characteristic
time scales is investigated. It is shown that even in simple
chaotic dynamical systems, the leading Lyapunov exponent is not
sufficient to estimate the predictability time.
This fact is due to the saturation of the error on the fast components
of the system which therefore do not contribute to the exponential growth
of the uncertainty at large error levels.
It is proposed to adopt a generalization of the Lyapunov exponent which
is based on the natural concept of error growing time at fixed error size.
The predictability time defined in terms of the Finite Size Lyapunov
Exponent displays a strong dependence on the error magnitude, as already
recognized by other authors.
The method is first illustrated on a simple numerical model obtained
by coupling two Lorenz systems with different time scales.
As a more realistic example, the analysis is then applied to a toy model
of the Atmospheric circulation recently introduced by Lorenz.
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| Dispersion of passive tracers in closed basins: beyond the
diffusion coefficient
|
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| V. Artale, G. Boffetta, A. Celani, M. Cencini and A. Vulpiani
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|
Physics of Fluids A 9, 3162-3171 (1997).
We investigate the spreading of passive tracers in closed basins.
If the characteristic length scale of the Eulerian velocities is
not very small compared with the size of the basin the usual
diffusion coefficient does not give any relevant information about
the mechanism of spreading.
We introduce a finite size characteristic time $\tau(\delta)$ which
describes the diffusive process at scale $\delta$.
When $\delta$ is small compared with the typical length of the velocity
field one has $\tau(\delta) \sim \lambda^{-1}$, where $\lambda$ is the maximum
Lyapunov exponent of the Lagrangian motion.
At large $\delta$ the behavior of $\tau(\delta)$ depends on the details
of the system, in particular the presence of boundaries, and in this limit
we have found a universal behavior for a large class of system under
rather general hypothesis.
The method of working at fixed scale $\delta$ makes more physical
sense than the traditional way of looking at the relative diffusion
at fixed delay times.
This technique is displayed in a series of numerical experiments
in simple flows.
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| Transient anomalous dispersion in random walkers
|
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| G. Boffetta, A. Celani and V. Rago
|
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|
Physics Letters A 235, 15-18 (1997).
A simple model of dispersive tracers which display
a transient anomalous regime is presented.
It is based on an ensemble of random walkers belonging
to two independent populations characterized by different
Lagrangian decorrelation times.
Apart from short-time ballistic and long-time diffusive behavior,
the dispersion shows anomalous scaling at intermediate times
over a wide range of variability for the free parameters of the model.
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| Predictability in Two Dimensional Decaying Turbulence
|
|
| G. Boffetta, A. Celani, A. Crisanti and A. Vulpiani
|
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|
Physics of Fluids A 9, 724-734 (1997).
Predictability problem for two-dimensional decaying turbulence is addressed by
means of numerical simulations. Qualitative and quantitative comparisons
with previous results obtained by closure approximations are critically
examined. It is found that, as for other features of two-dimensional
turbulence, the role of coherent vortices is essential for a correct
interpretation of the results.
A Lagrangian, vortex based, definition for the growth of incertitudes
leads in general to an enhancement of the predictability time.
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| Predictability in the large: an extension of the concept of
Lyapunov exponent
|
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| E. Aurell, G. Boffetta, A. Crisanti, G. Paladin and A. Vulpiani
|
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|
Journal of Physics A 30, 1-26 (1997).
We investigate the predictability problem in dynamical
systems with many degrees of freedom and a wide spectrum
of temporal scales. In particular, we study the case of
$3D$ turbulence at high Reynolds numbers by
introducing a finite-size Lyapunov exponent which measures the growth rate of
finite-size perturbations.
For sufficiently small perturbations this quantity coincides with
the usual Lyapunov exponent.
When the perturbation is still small compared to large-scale
fluctuations, but large compared to fluctuations at the
smallest dynamically active scales,
the finite-size Lyapunov exponent is inversely
proportional to the square of the perturbation size.
Our results are supported by numerical experiments on shell models.
We find that intermittency corrections do
not change the scaling law of predictability.
We also discuss the relation between
finite-size Lyapunov exponent and information entropy.
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| Growth of non-infinitesimal perturbations in turbulence
|
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| E. Aurell, G. Boffetta, A. Crisanti, G. Paladin e A. Vulpiani
|
|
|
Physical Review Letters 77, 1262-1265 (1996).
We discuss the effects of finite perturbations in fully developed turbulence
by introducing a measure of the chaoticity degree associated to a
given scale of the velocity field. This allows one to determine
the predictability time for non-infinitesimal perturbations,
generalizing the usual concept of maximum Lyapunov exponent.
We also determine the scaling law for our indicator in the framework
of the multifractal approach.
We find that the scaling exponent is not sensitive to intermittency
corrections, but is an invariant of the multifractal models.
A numerical test of the results is performed in the shell model for
the turbulent energy cascade.
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| Trapping of Passive Tracers in a Point Vortex System
|
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| G. Boffetta, A. Celani and P. Franzese
|
|
|
Journal of Physics A 29, 3749-3759 (1996).
The advection of passive markers in the flow generated by two point
vortices in a disk is studied.
This Hamiltonian dynamics is investigated by means of Poincar\'e
sections, via a set of appropriate canonical transformations.
As it is shown by n
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