Time irreversibility and multifractality of power along single particle trajectories in turbulence
Massimo Cencini Luca Biferale Guido Boffetta Massimo De Pietro
physics.flu-dyn cond-mat.stat-mech





 Corresponding author
massimo.cencini@cnr.it Istituto dei Sistemi
  Complessi, CNR, Via dei Taurini 19, 00185 Rome, Italy and INFN “Tor Vergata”


Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata”, Via Ricerca Scientifica 1, 00133 Roma, Italy


Dipartimento di Fisica and INFN, Università di Torino, 
Via P. Giuria 1, 10125 Torino, Italy



Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata”, Via Ricerca Scientifica 1, 00133 Roma, Italy



  The irreversible turbulent energy cascade epitomizes strongly
  non-equilibrium systems.  At the level of single fluid particles,
  time irreversibility is revealed by the asymmetry of the rate of
  kinetic energy change, the Lagrangian power, whose moments
  display a power-law dependence on the Reynolds number, as
  recently shown by Xu et al. [H Xu et al, Proc. Natl. Acad. Sci. U.S.A.
    111, 7558 (2014)].  Here Lagrangian power statistics are rationalized within the multifractal
  model of turbulence, whose predictions are shown to agree with
  numerical and empirical data.  Multifractal predictions are also
  tested, for very large Reynolds numbers, in dynamical models of the
  turbulent cascade, obtaining remarkably good agreement for
  statistical quantities insensitive to the asymmetry and,
  remarkably, deviations for those probing the asymmetry. These
  findings raise fundamental questions concerning time irreversibility
  in the infinite-Reynolds-number limit of the Navier-Stokes
  equations.


05.70.Ln,47.27.-i,47.27.eb



Time irreversibility and multifractality of power along single particle trajectories in turbulence[Version accepted for publication (postprint) on Phys. Rev. Fluids 2, 104604 – Published 27 October 2017]
    Massimo De Pietro
    
===========================================================================================================================================================================================================






§ INTRODUCTION


In nature, the majority of the processes
involving energy flow occur in nonequilibrium conditions from the
molecular scale of biology <cit.> to
astrophysics <cit.>. Understanding such nonequilibrium
processes is of great interest at both fundamental and applied levels,
from small-scale technology <cit.> to climate
dynamics <cit.>. A key aspect of nonequilibrium
systems is the behavior of fluctuations that markedly differ from
equilibrium ones.  As for the latter, detailed balance establishes
equiprobability of forward and backward transitions between any two
states, a statistical manifestation of time reversibility
<cit.>, while, irreversibility of nonequilibrium
processes breaks detailed balance.  In three-dimensional (3D)
turbulence, a prototype of very far-from-equilibrium systems,
detailed balance breaks in a fundamental way <cit.>: It is
more probable to transfer energy from large to small scales than its
reverse.  Indeed, in statistically stationary turbulence, energy,
supplied at scale L at rate ϵ (≈ U_L^3/L, U_L
being the root mean square single-point velocity), is transferred with
a constant flux approximately equal to ϵ up to the scale η, where it is
dissipated at the same rate ϵ, even for vanishing viscosity
(ν→ 0) <cit.>. As a result, time reversibility,
formally broken by the viscous term, is not restored for ν→ 0
<cit.>.  Time irreversibility is unveiled by the
asymmetry of two-point statistical observables. In particular, the
constancy of the energy flux directly implies, in the Eulerian frame,
a non vanishing third moment of longitudinal velocity difference
between two points at distance r (the 4/5 law
<cit.>) and, in the Lagrangian frame, a faster separation
of particle pairs backward than forward in time
<cit.>.

Remarkably, time irreversibility has been recently discovered
at the level of single-particle statistics <cit.> that
is not a priori sensitive to the existence of a nonzero energy
flux. This opens important challenges also at applied levels for
stochastic modelization of single-particle transport, e.g., in
turbulent environmental flows <cit.>.  Both experimental
and numerical data revealed that the temporal dynamics of Lagrangian
kinetic energy E(t)=1/2v^2(t), where v(t)= u(
x(t),t) is the Lagrangian velocity along a particle trajectory x(t), is characterized by events where E(t) grows slower than it 
decreases.  Such flight-crash events result in the asymmetry of
distribution of the Lagrangian power, p(t)=Ė= v(t)·
a(t) (a≡v̇=∂_t  u+ u·∇ u being the fluid particle acceleration). While in stationary
conditions the mean power vanishes ⟨ p⟩=0, the third
moment is increasingly negative with the Taylor scale
Reynolds number Re_λ≈ (U_LL/ν)^1/2≈
T_L/τ_η measuring the ratio between the timescales of energy
injection T_L and dissipation τ_η, which easily exceeds
10^3 in the laboratory.  In particular, it was found that ⟨
p^3⟩/ϵ^3 ∼ -Re_λ^2 <cit.> and
⟨ p^2⟩/ϵ^2 ∼Re_λ^4/3.
Interestingly, the Re_λ dependence deviates from the
dimensional prediction based on Kolmogorov phenomenology <cit.>
⟨ p^q⟩/ϵ^q ∝Re_λ^q/2, signaling that the Lagrangian power is strongly
intermittent as exemplified by its spatial distribution and the strong
non-Gaussian tails of the probability distribution function of p
(Fig. <ref>).





From a theoretical point of view, the above scaling behavior of the
power with Re_λ implies that the skewness of the probability
density function (PDF) of p, S = ⟨ p^3⟩ /⟨
p^2⟩ ^3/2 , is constant, suggesting that 
time irreversibility is robust and persists even in the limit
Re_λ→∞.  It is important to stress that one might use
different dimensionless measures of the symmetry breaking, e.g., S̃ = ⟨ p^3⟩ /⟨ |p|^3⟩ , which directly
probes the ratio between the symmetric and asymmetric contributions to
the PDF. In the presence of anomalous scaling S and S̃ can
have a different Re_λ dependence, as highlighted for the
problem of statistical recovery of isotropy <cit.>.

The aim of our work is twofold. First, we use direct numerical
simulations (DNSs) of 3D Navier-Stokes equations (NSEs) to quantify the
degree of recovery of time reversibility along single-particle
trajectories using different definitions as discussed above. Second,
we show that it is possible to extend the multifractal formalism (MF)
<cit.> to predict the scaling of the absolute value of the
Lagrangian power statistics.  Moreover, in order to explore a wider range of
Reynolds numbers, we also investigate the equivalent of the Lagrangian
power statistics in shell models <cit.>.

The rest of the paper is organized as follows.
Section <ref> is devoted to a brief review of the multifractal formalism
for fully developed turbulence and the predictions for the statistics 
of the Lagrangian power. In Sec. <ref> we compare these 
predictions with the results obtained from direct numerical simulations
of the Navier-Stokes equations and from a shell model of turbulence.
Section <ref> is devoted to a summary and conclusions. The Appendix
reports some details of the numerical simulations.




§ THEORETICAL PREDICTIONS BY THE MULTIFRACTAL MODEL


We start by recalling the MF for the
Eulerian statistics <cit.>. The basic idea is to
replace the global scale invariance in the manner of Kolmogorov 
with a local scale invariance, by assuming that spatial velocity
increments δ_r u over a distance r≪ L are characterized by
a range of scaling exponents h∈ℐ≡(h_m,h_M),
i.e., δ_r u∼ u_L (r/L)^h.  Eulerian structure functions
⟨ (δ_r u)^q⟩ are obtained by integrating over
h∈ℐ and the large-scale velocity u_L statistics
𝒫(u_L), which can be assumed to be independent of h. The MF
assumes the exponent h to be realized on a fractal set of dimension
D(h), so the probability to observe a particular value of h,
for r≪ L, is 𝒫_h(r) ∼ (r/L)^3-D(h). Hence, we find
⟨ (δ_r u)^q⟩∼⟨ u_L^q⟩∫_h∈ℐ dh (r/L)^hq+3-D(h)∼⟨ u_L^q⟩
(r/L)^ζ_q, where a saddle-point approximation for r≪ L
gives

    ζ_q= inf_h∈ℐ{hq+3-D(h)} .

For the MF to be predictive, D(h) should be
derived from the NSE, which is out of reach. One can, however,
use the measured exponents ζ_p and, by inverting
(<ref>), derive an empirical D(h). 
Here, following <cit.>, we use

    D(h)= 3-d_0-d(h)[ln(d(h)/d_0) -1] ,

with d(h)=3(1/9-h)/lnβ and d_0=2/[3(1-β)] corresponding,
via (<ref>), to
ζ_q=q/9+(2/3)(1-β^q/3)/(1-β), which, for β=0.6,
fits measured exponents fairly well <cit.>.

The MF has been extended from Eulerian to Lagrangian velocity
increments <cit.>.  The idea is that temporal
velocity differences δ_τ v over a time lag τ, along
fluid particle trajectories, can be connected to equal time spatial
velocity differences δ_r u by assuming that the largest
contribution to δ_τ v comes from eddies at a scale r such
that τ∼ r/δ_r u.  This implies δ_τ v ∼δ_r u, with

    τ∼ T_L (r/L)^1-h  ,

where T_L=L/u_L.  By combining Eq. (<ref>) and the D(h) obtained
from Eulerian statistics, one can derive a prediction for Lagrangian
structure functions, which has been found to agree with experimental
and DNS data
<cit.>.
The MF can be used also for describing the statistics of the 
acceleration a along fluid elements <cit.>.
The acceleration can be estimated by assuming 

    a ∼δ_τ_η v/τ_η .

According to the MF, the dissipative
scale fluctuates as η∼ (ν L^h/u_L)^1/(1+h) 
<cit.>, which leads, via (<ref>), to

    τ_η∼ T (ν/Lu_L)^(1-h)/(1+h) .

Substituting (<ref>) in
(<ref>) yields the acceleration conditioned on given values of
h and u_L:

    a ∼ν^(2h-1)/(1+h) u_L^3/(1+h) L^-3h/(1+h) .

Equation (<ref>) has been successfully used to predict the
acceleration variance <cit.> and PDF <cit.>.

We now use
(<ref>) to predict the scaling behavior of the Lagrangian
power moments with Re_λ.  These can be estimated as ⟨
p^q ⟩∼⟨ (au_L)^q⟩∼∫ du_L
𝒫(u_L) ∫_h∈ℐ dh 𝒫_h(τ_η)
(au_L)^q with
𝒫_h(τ_η)=(τ_η/T)^[3-D(h)]/(1-h).  Using
(<ref>) with ν=U_LL Re_λ^2 (with
U_L^2=⟨ u_L^2 ⟩), we have

    ⟨ p^q⟩/ϵ^q∼∫dṽ𝒫(ṽ)∫_h∈ℐdh  ṽ^[4q+h-3+D(h)]/(1+h)Re_λ^2 [(1-2h)q-3+D(h)]/(1+h) ,

with ṽ=u_L/U_L
[Possible divergences in ṽ→ 0 should not be a concern as
  the MF cannot be trusted for small velocities].  In the limit
Re_λ→∞, a saddle point approximation of the integral
(<ref>) yields, up to a multiplicative constant (depending on
the large scale statistics), ⟨ p^q⟩/ϵ^q∼Re_λ^α(q) with

    α(q)=sup_h{2(1-2h)q-3+D(h)/1+h} .





§ COMPARISON WITH NUMERICAL SIMULATIONS




To test the MF predictions (<ref>) we use two sets of DNS of
homogeneous isotropic turbulence on cubic lattices of sizes from
128^3 up to 2048^3, with Re_λ up to 540, obtained with
two different forcings (see the Appendix for
details). In particular, to probe both the symmetric and asymmetric
components of the Lagrangian power statistics, we study the nondimensional moments

    𝒮_q=⟨
    |p|^q⟩/ϵ^q,   𝒜_q=⟨
    p|p|^q-1⟩/ϵ^q ,

where the latter vanishes for a symmetric (time-reversible) PDF. In
Fig. <ref> we show the second-and third-order moments of
(<ref>) as a function of Re_λ. 
We observe that (i) the MF prediction
(<ref>) is in excellent agreement with the scaling of
𝒮_q (see also Fig. <ref>) and (ii) the asymmetry
probing moments 𝒜_q are negative, confirming the existence
of the time-symmetry breaking, and scale with exponents compatible
with those of 𝒮_q. This implies that time reversibility is
not recovered even for Re_λ→∞.  Actually,
irreversibility is independent of Re_λ if measured in terms of
the homogeneous asymmetry ratio S̃=𝒜_q/𝒮_q, while if quantified in
terms of the standard skewness S, it grows as Re_λ^χ
with χ=α(3)-(3/2) α(2) ≃ 0.35 due to anomalous scaling. 
In the
inset of Fig. <ref> we compare S with S̃.
Evaluating (<ref>) with D(h) given by (<ref>), we
obtain α(2)≈ 1.17 and α(3)≈ 2.10, which
are close to the 4/3 and 2 reported in <cit.>. We remark that
the authors of <cit.> explained the observed exponents by
assuming that the dominating events are those for which the particle
travels a distance r ∼ U_Lτ in a frozenlike turbulent
velocity field, so that δ_τ_η v ∼ (ϵτ_η U_L)^1/3. Hence, for the acceleration
(<ref>) one has a ∼
U_L^1/3ϵ^1/3τ_η^-2/3, which, using the
dimensional prediction τ_η=(ν/ϵ)^1/2, ends up
in p∼ U_L a∼
U_L^4/3ϵ^2/3ν^-1/3∼ϵRe_λ^2/3. This argument provides only a linear
approximation 2q/3 for α(q), while the multifractal model
is able to describe its nonlinear dependence on q. In
Fig. <ref> we show the whole set of exponents for both
𝒜_q and 𝒮_q as observed in DNS data and 
compare them with the prediction (<ref>).


It is worth noticing that the MF provides an excellent prediction for the
statistics of p also in 1D compressible turbulence, i.e., in the
Burgers equation, studied in <cit.>. Here, out of a smooth
(h=1) velocity field, the statistically dominant structures are
shocks (h=0). The velocity statistics is thus bifractal with
D(1)=1 and D(0)=0 <cit.>.  Adapting (<ref>) to one dimension
and noticing that Re∝Re_λ^2, we have ⟨
p^q⟩∼Re^α_1D(q) with
α_1D(q)=sup_h{[(1-2h)q-1+D(h)]/(1+h)}, which for Burgers
means α_1D(q)=q-1, in agreement with the results of
<cit.>.

To further investigate the scaling behavior of
the symmetric and asymmetric components of the power statistics
in a wider range of  Reynolds numbers and with higher statistics,
in the following we study Lagrangian power  within the framework of shell
models of turbulence <cit.>.
Shell models are dynamical systems built to reproduce the basic phenomenology of
the energy cascade on a discrete set of scales, r_n=k_n^-1=L2^-n
(n=0,…,N), which allow us to reach high Reynolds numbers.  For
each scale r_n, the velocity fluctuation is represented by a single
complex variable u_n, which evolves according to the differential equation <cit.>

    u̇_n=ik_n(u_n+2u^*_n+1-1/4u_n+1u^*_n-1
    +1/8u_n-1u_n-2)-ν k_n^2 u_n+f_n

whose structure is a cartoon of the 3D NSE in Fourier space but for the
nonlinear term that restricts the interactions to neighboring shells,
as justified by the idea localness of the energy cascade
<cit.>.  Energy is injected with rate
ϵ=⟨∑_n Re{f_n u_n^*}⟩.
See the Appendix for details on forcing
and simulations. As shown in <cit.>, this model displays anomalous scaling for
the velocity structure functions, ⟨ |u_n|^q⟩∼
k_n^-ζ_q, with exponents remarkably close to those observed in
turbulence and in very good agreement with the MF prediction
(<ref>).

Following <cit.>, we model the Lagrangian velocity along
a fluid particle as the sum of the real part of velocity fluctuations
at all shells v(t)≡∑_n=1^NRe{u_n}.  Analogously, we
define the Lagrangian acceleration a≡∑_n=1^NRe{u̇_n} and power p(t)=v(t)a(t).

In Figs. <ref>(a)  and <ref>(b) we show the moments 𝒮_q and
𝒜_q for q=2,3 obtained from the shell model. The
symmetric ones 𝒮_q perfectly agree with the multifractal
prediction obtained using the same D(h), i.e., (<ref>) for
β=0.6, which fits the Eulerian statistics. The
asymmetry-sensitive moments 𝒜_q are negative (for q>1),
as in Navier-Stokes turbulence, and display a power-law dependence on
Re_λ with a different scaling respect to their symmetric analogs. In
particular, as summarized in Fig. <ref>(c), we observe smaller
exponents with respect to the MF up to q=4. Rephrased in terms of the
skewness, these findings mean that the time asymmetry becomes weaker
and weaker with increasing Reynolds numbers if measured in terms of
S̃ [Fig. <ref>(c) inset], as distinct from what was observed for the NSE (Fig. <ref> inset). The standard
skewness S, on the other hand, is still an increasing function of
Re_λ though with an exponent smaller than the
MF prediction α(3)-(3/2)α(2), because 𝒜_3 has a
shallower slope  than the multifractal one.



§ CONCLUSIONS


We have shown that the
multifractal formalism predicts the scaling behavior of the Lagrangian
power moments in excellent agreement with DNS data and with previous
results on the Burgers equation. In the range of explored
Re_λ, we have found that symmetric and antisymmetric moments
share the same scaling exponents, and therefore the MF is able to
reproduce both statistics.  It is worth stressing that the effectiveness of the MF in
describing the scaling of 𝒜_q is not obvious  as the
MF, in principle, bears no information on statistical
asymmetries [See Sect.8.5.4 in <cit.> for a
  discussion.]. By analyzing the Lagrangian power
statistics in a shell model of turbulence, at Reynolds numbers much
higher than those achievable in DNS, we found that symmetric and
antisymmetric moments possess two different sets of exponents.  While
the former are still well described by the MF formalism, the latter,
in the range of q explored, are smaller.  As a consequence, the
ratios 𝒜_q/𝒮_q in the shell model decrease with
Re_λ. However, we observe that the mismatch between the two
sets of scaling is compatible with the assumption that 𝒜_q
∼𝒮_q ⟨sign(p)⟩, i.e., that the main effect
is given by a cancellation exponent introduced by the scaling of sgn(p). Our findings raise the question whether the apparent
similar scaling among symmetric and asymmetric components in the NSE is
robust for large Reynolds numbers or a sort of recovery of
time symmetry would be observed also in Navier-Stokes turbulence as
for shell models.


We conclude by mentioning another interesting open question. In
<cit.> it was found that the Lagrangian power
statistics is asymmetric also in statistically stationary 2D
turbulence in the presence of an inverse cascade. Like in three dimensions,
the third moment is negative and its
magnitude grows with the separation between the timescale
of dissipation by friction (at large scale) and of energy
injection (at small scale), which is a measure of Re_λ
for the inverse cascade range. Moreover, the scaling exponents are
quantitatively close to the 3D ones. This raises the question on the
origin of the scaling in 2two dimensions that cannot be rationalized within the MF,
since the inverse cascade is not intermittent
<cit.>. Likely, to answer the question one needs a better
understanding of the influence of the physics at and below the forcing
scale on the 2D Lagrangian power.




We thank F. Bonaccorso for computational support.  We acknowledge
support from the COST Action MP1305 “Flowing Matter.”  L.B.
 and M.D.P. acknowledge funding from ERC under the EU 7^th Framework
 Programme, ERC Grant Agreement No. 339032. G.B. acknowledge
 Cineca within the INFN-Cineca agreement INF17­fldturb.



*


§ DETAILS ON THE NUMERICAL SIMULATIONS





 §.§ Direct Numerical Simulations

We performed two sets of DNSs at different 
resolutions and Reynolds numbers with two different forcing schemes.
The values of the parameters characterizing all the simulations are shown in Table <ref>.
In all cases we integrated the Navier-Stokes equations

    ∂_t  u  +  u·∇ u≡ a 
    = - ∇ P + νΔ u  +  f ,

for the incompressible velocity field u( x,t)
with a fully parallel pseudo-spectral code, fully dealiased with 2/3 rule <cit.>, in a cubic box 
of size ℒ=2 π with periodic boundary conditions.
In (<ref>) P represents the pressure and
ν is the kinematic viscosity of the fluid.

For the set of runs DNS1 we used a Sawford-type stochastic
forcing, involving the solution of the stochastic differential
equations <cit.>

    d f̃_i = ã_i(t) dt   ,
    
    d ã_i = -a_1 ã_i(t) dt -a_2 f̃_i(t) dt + a_3 dW_i(t)  ,

where a_1 = 1/τ_f, a_2 = (1/8)/τ_f^2, a_3 = √(2 a_1
  a_2), and dW_i(t) = r √(dt) is an increment of a Wiener
process (r is a random Gaussian number with ⟨ r ⟩ = 0
and ⟨ r^2 ⟩ = 1). The forcing f(k,t) in
Fourier space is then

    f(k,t) = 
    
    i k× [i k× (0.16   k^-4/3f̃)]      for   k ∈ [k_f,min, k_f,max]  
    
    0      for   k ∉ [k_f,min, k_f,max]   .

Time integration is performed by a second-order Adams-Basforth scheme with exact integration of the linear dissipative term <cit.>.




For the set of runs DNS2 we use a deterministic forcing acting
on a spherical shell of wavenumbers in Fourier space 
0 < | k| ≤ k_f, where 
k_f=1.5 with imposed energy input rate ε
<cit.>. 
In Fourier space the forcing reads

    f(k,t) = 
    εu( k,t) /[2 E_f(t)]      for   k ∈ [k_f,min, k_f,max]  
    
    0      for   k ∉ [k_f,min, k_f,max]   .

where E_f(t)=∑_k=0^k_f E(k,t), and E(k,t) is the energy
spectrum at time t. This forcing guarantees the constancy of the
energy injection rate. Notice that Eq. (<ref>) explicitly breaks
the time-reversal symmetry; however, owing to the universality
properties of turbulence with respect to the forcing, we expect this
effect to be negligible as compared to the energy cascade.  Time
integration is performed by a second-order Runge-Kutta midpoint method
with exact integration of the linear dissipative term
<cit.>.  Simulations have a
resolution N sufficient to resolve the dissipative scale with
k_maxη≃ 1.7 (k_max=N/3).  We have checked in the
simulations that the velocity field is statistically isotropic with a
probability density function (for each component) close to a Gaussian.

Simulations are performed for several large-scale eddy turnover times T,
after an initial transient to reach the turbulent state, in
order to generate independent velocity fields in stationary conditions. 
From the velocity fields 
the acceleration field is then computed by evaluating the right hand side of (<ref>)
and the power field is obtained as p= u· a.




 §.§ Simulations of the shell model


As for the shell model (<ref>), simulations have been
performed by fixing the number of shells N=30 and varying the
viscosity ν in the range [3.16 × 10^-4 , 3.16 ×
  10^-8]. For each value of ν we performed ten independent
realizations lasting approximately 10^6 T_L each.  Time integration is
performed using a fourth-order Runge-Kutta scheme with exact
integration of the linear term.  Forcing is stochastic and acts only
on the first shell f_n=fδ_n,1.  The stochastic forcing is
obtained by choosing f=F(f^R+if^I) with F=1 and

    ḟ^α   =    -1/τ_ff^α + √(2)/τ_fθ^α(t)   ,
    θ̇^α   =    -1/τ_fθ^α + √(2/τ_f)η^α(t)   ,

where η^α is a zero mean Gaussian variable with correlation
⟨η^α(t)η^β(t')⟩ =
δ_αβδ(t-t'). As a result, f^α is a zero
mean Gaussian variable with correlation ⟨
f^α(t)f^β(t')⟩= δ_αβ1/τ_fexp(-|t-t'|/τ_f) (|t-t'|+τ_f).  In particular,
we used τ_f=1, which is of the order of the large-eddy turnover
time T_L.  Using a constant amplitude forcing, we obtained, within
error bars, indistinguishable exponents (not shown).










35
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Ritort(2008)]ritort2008nonequilibrium
  author author F. Ritort, title title Nonequilibrium
  fluctuations in small systems: From physics to biology, @noop 
  journal journal Adv. Chem. Phys. volume 137, pages 31 (year
  2008)NoStop
[Priest(2014)]priest2012solar
  author author E. Priest, @noop  title Magnetohydrodynamics of
  the Sun (publisher Cambridge University Press, Cambridge, year 2014)NoStop
[Blickle and Bechinger(2012)]blickle2012realization
  author author V. Blickle and author C. Bechinger, title title Realization
  of a micrometre-sized stochastic heat engine, @noop  journal journal Nature Phys. volume
  8, pages 143 (year 2012)NoStop
[Kleidon(2010)]kleidon2010life
  author author A. Kleidon, title title Life,
  hierarchy, and the thermodynamic machinery of planet earth, @noop 
  journal journal Phys. Life Rev. volume 7, pages 424–460 (year
  2010)NoStop
[Onsager(1931)]onsager1931
  author author L. Onsager, title title Reciprocal
  relations in irreversible processes. I. @noop  journal journal Phys. Rev. volume
  37, pages 405 (year 1931)NoStop
[Rose and Sulem(1978)]rose1978
  author author H. A. Rose and author P. L. Sulem, title title Fully developed
  turbulence and statistical mechanics, @noop  journal
  journal J. Phys-Paris volume 39,  pages 441–484 (year 1978)NoStop
[Frisch(1995)]Frisch1995
  author author U. Frisch, @noop  title Turbulence: The Legacy
  of A.N. Kolmogorov (publisher Cambridge University Press, Cambridge,  year 1995)NoStop
[Falkovich and Sreenivasan(2006)]falkovich2006
  author author G. Falkovich and author K. R. Sreenivasan, title title Lessons
  from hydrodynamic turbulence, @noop  journal
  journal Phys. Today volume 59,  pages 43 (year 2006)NoStop
[Falkovich and Frishman(2013)]falkovich2013single
  author author G. Falkovich and author A. Frishman, title title Single flow
  snapshot reveals the future and the past of pairs of particles in
  turbulence, @noop  journal journal
  Phys. Rev. Lett. volume 110, pages
  214502 (year 2013)NoStop
[Jucha et al.(2014)Jucha,
  Xu, Pumir, and Bodenschatz]jucha2014time
  author author J. Jucha, author H. Xu, author A. Pumir, and author
  E. Bodenschatz, title
  title Time-reversal-symmetry breaking in
  turbulence, @noop  journal journal
  Phy. Rev. Lett. volume 113, pages
  054501 (year 2014)NoStop
[Xu et al.(2014)Xu,
  Pumir, Falkovich, Bodenschatz, Shats, Xia, Francois, and Boffetta]xu2014
  author author H. Xu, author A. Pumir, author G. Falkovich, author
  E. Bodenschatz, author
  M. Shats, author H. Xia, author N. Francois, and author G. Boffetta, title title Flight–crash
  events in turbulence, @noop  journal journal Proc. Natl. Acad. Sci. volume 111,  pages 7558–7563 (year 2014)NoStop
[Pumir et al.(2014)Pumir,
  Xu, Boffetta, Falkovich,  and Bodenschatz]xu2014b
  author author A. Pumir, author H. Xu, author G. Boffetta, author
  G. Falkovich, and author
  E. Bodenschatz, title
  title Redistribution of kinetic energy in turbulent
  flows, @noop  journal journal Phys.
  Rev. X volume 4, pages 041006
  (year 2014)NoStop
[Wilson and Sawford(1996)]sawford
  author author J. D. Wilson and author B. L. Sawford, title title Review of
  Lagrangian stochastic models for trajectories in the turbulent atmosphere,  @noop  journal journal Bound.-Layer
  Meteor. volume 78, pages 191–210
  (year 1996)NoStop
[Biferale and Vergassola(2001)]BV2001
  author author L. Biferale and author M. Vergassola, title title Isotropy vs
  anisotropy in small-scale turbulence, @noop  journal
  journal Phys. Fluids volume 13,  pages 2139–2141 (year 2001)NoStop
[Frisch and Parisi(1985)]FP1985
  author author U Frisch and author G Parisi,  in @noop  booktitle Turbulence and Predictability in Geophysical Fluid Dynamics and
  Climate Dynamics, editor edited by editor
  U. M. Ghil (publisher
  North-Holland, Amsterdam, year 1985)NoStop
[Biferale(2003)]biferale2003
  author author L. Biferale, title title Shell models
  of energy cascade in turbulence, @noop  journal
  journal Annu. Rev. Fluid Mech. volume
  35, pages 441–468 (year 2003)NoStop
[Bohr et al.(2005)Bohr,
  Jensen, Paladin, and Vulpiani]bohr2005
  author author T. Bohr, author M. H. Jensen,
  author G. Paladin, and  author A. Vulpiani, @noop
   title Dynamical Systems Approach to Turbulence  (publisher Cambridge University Press, Cambridge, year
  2005)NoStop
[She and Leveque(1994)]SL1994
  author author Z.-S. She and author E. Leveque,  title title Universal scaling laws in
  fully developed turbulence, @noop  journal journal Phys. Rev. Lett. volume 72, pages 336 (year 1994)NoStop
[Arnéodo et al.(2008)Arnéodo et al.]arneodo2008
  author author A. Arnéodo et al., title title Universal intermittent properties of particle trajectories in highly
  turbulent flows, @noop  journal journal
  Phys. Rev. Lett. volume 100, pages
  254504 (year 2008)NoStop
[Borgas(1993)]borgas1993
  author author M. S. Borgas, title title The multifractal
  lagrangian nature of turbulence, @noop  journal
  journal Philos. Trans. volume 342,  pages 379–411 (year 1993)NoStop
[Boffetta et al.(2002)Boffetta, De Lillo, and Musacchio]boffetta2002
  author author G. Boffetta, author F. De Lillo, and author S. Musacchio,  title title Lagrangian statistics and
  temporal intermittency in a shell model of turbulence, @noop 
  journal journal Phys. Rev. E volume 66, pages 066307 (year
  2002)NoStop
[Chevillard et al.(2003)Chevillard, Roux, Lévêque,
  Mordant, Pinton, and Arnéodo]chevillard2003
  author author L. Chevillard, author S. G. Roux, author E. Lévêque, author N. Mordant, author J.-F. Pinton, and author A. Arnéodo,  title title Lagrangian velocity
  statistics in turbulent flows: Effects of dissipation, @noop 
  journal journal Phys. Rev. Lett. volume 91, pages 214502 (year
  2003)NoStop
[Biferale et al.(2004)Biferale, Boffetta, Celani, Devenish, Lanotte, and Toschi]biferale2004
  author author L. Biferale, author G. Boffetta,
  author A. Celani, author B. J. Devenish, author
  A. S. Lanotte, and author
  F. Toschi, title title Multifractal statistics of lagrangian velocity and
  acceleration in turbulence, @noop  journal journal Phys. Rev. Lett. volume 93, pages 064502 (year 2004)NoStop
[Frisch and Vergassola(1991)]FV1991
  author author U. Frisch and author M. Vergassola, title title A prediction
  of the multifractal model: the intermediate dissipation range, @noop
   journal journal Europhys. Lett. volume 14, pages 439 (year 1991)NoStop
[Note1()]Note1
  note Possible divergences in tilde07Ev→ 0 should not be a concern as the MF cannot be trusted for
  small velocitiesNoStop
[Grafke et al.(2015)Grafke,
  Frishman, and Falkovich]grafke2015
  author author T. Grafke, author A. Frishman,
  and author G. Falkovich,  title title Time irreversibility of the
  statistics of a single particle in compressible turbulence, @noop 
  journal journal Phys. Rev. E volume 91, pages 043022 (year
  2015)NoStop
[Bec and Khanin(2007)]bec2007
  author author J. Bec and author K. Khanin,  title title Burgers turbulence,  @noop  journal journal Phys. Rep.  volume 447, pages 1–66 (year 2007)NoStop
[L’vov et al.(1998)L’vov, Podivilov, Pomyalov, Procaccia, and Vandembroucq]Itamar
  author author V. S. Lvov, author E. Podivilov,
  author A. Pomyalov, author I. Procaccia, and author D. Vandembroucq, title title Improved shell model of turbulence,  @noop  journal journal Phys. Rev. E  volume 58, pages 1811 (year
  1998)NoStop
[Note2()]Note2
  note See Sect.8.5.4 in <cit.> for a
  discussion.Stop
[Boffetta and Ecke(2012)]boffetta2012
  author author G. Boffetta and author R. E. Ecke, title title Two-dimensional
  turbulence, @noop  journal journal
  Annu. Rev. Fluid Mech. volume 44, pages 427–451 (year 2012)NoStop
[Orszag(1971)]orszag1971elimination
  author author S. A. Orszag, title title On the
  elimination of aliasing in finite-difference schemes by filtering
  high-wavenumber components, @noop  journal journal J. Atmos. Sciences volume 28, pages 1074–1074 (year 1971)NoStop
[Sawford(1991)]sawford1991
  author author B. L. Sawford, title title Reynolds number
  effects in Lagrangian stochastic models of turbulent dispersion,  @noop  journal journal Phys. Fluids A  volume 3, pages 1577–1586 (year 1991)NoStop
[Canuto and Quarteroni(2006)]canuto2006spectral
  author author C. Canuto and author A. Quarteroni, @noop  title Spectral Methods  (publisher Wiley Online Library, New York, year
  2006)NoStop
[Lamorgese et al.(2005)Lamorgese, Caughey, and Pope]lamorgese2005direct
  author author A. G. Lamorgese, author D. A. Caughey, and author S. B. Pope, title title Direct numerical
  simulation of homogeneous turbulence with hyperviscosity, @noop 
  journal journal Phys. Fluids volume 17, pages 015106 (year
  2005)NoStop
[Boyd(2001)]boyd2001chebyshev
  author author J. P. Boyd, @noop  title Chebyshev and Fourier
  Spectral Methods (publisher Courier, North Chelmsford, year 2001)NoStop




