Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers Equation

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by  Michele Buzzicotti

Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers equation ∗
Michele Buzzicotti,1,† Luca Biferale,1,‡ Uriel Frisch,2,§ and Samriddhi Sankar Ray3,¶
1Department of Physics and INFN, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome, Italy.
2Lab. Lagrange, UCA, OCA, CNRS, CS, 34229, 06304, Nice Cedex 4, France
3International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
Wepresenttheoreticalandnumericalresultsfortheone-dimensionalstochasticallyforcedBurgers
equation decimated on a fractal Fourier set of dimension D.
We investigate the robustness of the energy transfer mechanism and of the small-scale statistical
fluctuations by changing D. We find that a very small percentage of mode-reduction (D (cid:46) 1) is
enoughtodestroymostofthecharacteristicsoftheoriginalnon-decimatedequation. Inparticular,
6 weobserveasuppressionofintermittentfluctuationsforD<1andaquasi-singulartransitionfrom
1 the fully intermittent (D = 1) to the non-intermittent case for D (cid:46) 1. Our results indicate that
0 the existence of strong localized structures (shocks) in the one-dimensional Burgers equation is the
2 result of highly entangled correlations amongst all Fourier modes.

r
a PACSnumbers: 47.27.Gs,05.20.Jj
M

1 I. INTRODUCTION cascade in physical space [1]. These contrast with other
2 attempts, based on the spectral space, involving statis-
tical closures and renormalization group methods. De-
] An outstanding challenge of the past few decades has
spitetheirsuccessforcertainproblems, noneoftheseat-
n been to develop a rigorous understanding of the energy
y tempts have been able to quantitatively connect anoma-
transfer from large to small scales in fully developed,
d lous scaling with the structure of the original equation,
three-dimensional, incompressible turbulent flows [1].
- and hence with its intermittent behavior. As a result,
u Numericalsimulationsandexperimentsshowthatmulti-
our understanding of anomalous scaling is still based on
l point velocity correlation functions are intermittent, i.e.,
f phenomenological real-space descriptions and real-space
. theydevelopapower-lawbehaviorwithnon-dimensional
s methodologies.
c (anomalous)scalingexponents[1–3]. Thequestionofthe
i originsofintermittencyinturbulenceanditsrelationin- Intermittency is intimately connected with ideas of
s
energy transfer from large to small scales. Working in
y ter alia to small structures is also of central importance
h in the areas of non-equilibrium statistical physics, fluid Fourier space should, hopefully, open new and possible
p ways to understand this cascade. In a recent work [18],
dynamics, astrophysics, and geophysics [4–15].
[ the idea of fractal decimation was introduced, with the
In this paper, we investigate the small-scale proper-
aim of studying the evolution of the Navier-Stokes equa-
2
ties of the stochastically-forced one-dimensional Burgers
tions on an effective dimension D out of an integer d-
v
equationwhich,isaparadigmaticexampleofahighlyin-
7 dimensional embedding manifold. This is done by in-
termittent system with statistics dominated by coherent
9 troducing a quenched mode-reduction in Fourier space
6 structures in physical space (shocks) [16, 17]. For this, such that in a sphere of radius k the number of modes
3 weperformaseriesofnumericalexperimentsbystudying that are involved in the dynamics scale as kD (where
0 the evolution of the original partial differential equation
D < d is the effective fractal dimension of the system)
. restrictedonafractalsetofFouriermodes[18]. Theidea
1 for large k [18, 19]. This approach allows us to decimate
0 is to reduce the number of degrees of freedom with min- the number of triad interactions in Fourier space as a
6 imal breaking of the original symmetries of the system.
function of the wavenumbers involved as well as to con-
1 The goal is to understand what are the key ingredients
sider the problem in non-integer, fractal dimensions D.
v: in the dynamics necessary to reproduce the main sta- In Ref.[20] the first results for a set of simulations of the
i tistical properties of the original non-decimated Burgers decimated, three-dimensional (3d) Navier-Stokes equa-
X
equationandthereforetounderstandtherobustnessand
tion have been reported with the intriguing conclusion
r origins of its shock-like structures.
a thatfractalFourierdecimationleads,ratherquickly(i.e.,
Over the past few decades various models of intermit- for a very small reduction of the Fourier modes D (cid:46) 3),
tency have been developed based on the idea of energy to vanishing intermittency.
In the present paper we investigate the same problem
for the one-dimensional Burgers equation. The main ad-
vantage with respect to the previous attempt on the 3D
∗Postprint version of the article published on PHYSICAL RE- Navier-Stokes is that here, due to the simpler structure
VIEWE93,033109(2016)
of the problem, numerical simulations can reach much
†Electronicaddress: [email protected]
higher resolutions and therefore assess, in a fully quan-
‡Electronicaddress: [email protected]
§Electronicaddress: [email protected] titative way, the problem of scaling and corrections to
¶Electronicaddress: [email protected] it.
2
Therestofthepaperisorganisedasfollows. InSec.II WeperformedasetofnumericalsimulationsofEq.(3)
we introduce the decimated Burgers equation and give by changing the dimension D between 0.70 D 1.0
detailsaboutournumericalsimulations. Wethenpresent and the number of collocation points N from≤216 t≤o 219.
resultsfromournumericalsimulationsaswellasprovide WechoosetheforcingtobeGaussianandwhite-in-time,
theoretical and phenomenological arguments to substan- such that fˆ(k ,t )fˆ(k ,t ) = f k −1δ(t t )δ(k +
1 1 2 2 0 f 1 2 1
(cid:104) (cid:105) | | −
tiatethem. Inparticular,inSec.IIIweexaminetheeffect k ),actingonlyonashellofwavenumbersatlargescales
2
ofdecimationonsecond-ordercorrelationfunctionsboth k [1 : 5 10]. We use a pseudo-spectral method
f
∈ −
in Fourier space, via the energy spectra (Sec. IIIA), and with a second-order Adams-Bashforth scheme to inte-
in physical space (Sec. IIIB) through the second-order grate in time. Details of all simulations can be found
structure function and flatness. We then, in Sec. IV, in- in Table I. We note that the values of ν decreases with
vestigate in detail – by using numerical simulations and the dimension D (see Table I); indeed as the decimation
theory – the suppression of intermittency by examining becomes stronger the contribution of the non-linear ad-
thehigher-orderstructurefunctions. FinallyinSec.Vwe vection term becomes weaker [18]. Hence to compare
make concluding remarks and provide a plausible theo- results from simulations with different values of D, we
retical framework in which to understand the spectral use smaller and smaller values of ν in order to observe a
scaling seen in our simulations. similar extension of the inertial range. We do not know
if the decimated equations are well behaved as ν 0
→
and if the system develops a dissipative anomaly leading
II. THE BURGERS EQUATION ON A to a stationary behavior for all D. This is an interesting
FRACTALLY DECIMATED FOURIER SET
point that will be addressed in future work.
Hereafter we analyze statistical properties for either
Following Ref. [18] we define the fractal Fourier dec- a single quenched realization of the decimation mask or
imation operator PD acting on a generic field u(x,t) = afterafurtheraveragingoverdifferentrealizationsofthe
(cid:80)keikxuˆk(t) as: quencheddisorder. Weindicatewith(cid:104)•(cid:105)theaverageover
timeforasinglerealizationofthedecimationmask;while
(cid:88)
v(x,t)=P u(x,t)= eikxθ uˆ (t) , (1) is used to denote an average over different quenched
D k k
•
k masks, where each mask acts as a projector.
ItiswellknownthattheordinaryforcedBurgersequa-
where θ are independently chosen random numbers,
k tion develops several discontinuities (shocks) connected
withθ =θ ,suchthatθ =1,withprobabilityh and
k −k k k by smooth, continuous ramps as it evolves in time [see
θ =0, withprobability1 h . Bychoosingh =kD−1,
k k k Fig. (1a)]. As soon as we introduce the fractal deci-
−
with 0 < D 1, we introduce a quenched disorder that
mationprojector,severalsharposcillatorystructuresap-
≤
suppresses, randomly, modes on the Fourier lattice. On
pear in the solution for v(x,t), even for mild decimation
average we have N(k) kD surviving modes at a dis- (D (cid:46) 1), as seen in Fig 1(b). Such structures, although
∝
tance k from the origin. Considering u as the velocity
reminiscentoffeatures(tygers)oftheGalerkin-truncated
field given by the solution of the forced, one-dimensional
Burgersequation[22],arecruciallydifferentbecausethey
Burgers equation:
are spatially much more delocalized.
∂u 1∂u2 ∂2u
+ =ν +f, (2)
∂t 2 ∂x ∂x2
III. SECOND-ORDER FUNCTIONS
where ν is the viscosity, u is 2π space-periodic in x and
f is a stochastic force acting on a few shells which drives A. Scaling in Fourier Space: The Energy Spectra
the system to a statistical steady state. We can then
write the decimated Burgers equation, which gives the The first question we want to address is the effect of
evolution for the decimated field v as: decimation on the mean spectral properties. We define
the energy spectrum for a general fractal dimension as
∂v 1 ∂v2 ∂2v
∂t + 2PD ∂x =ν∂x2 +f, (3) Ek = θk(cid:104)uˆ2k(cid:105). Since fractal decimation does not break
scaling invariance of the original equation in the inviscid
with initial conditions v = P u [21]. When D = 1 limit, we still expect to observe a power-law dependency
0 D 0
we recover the usual one-dimensional equation. It is im- as a function of the wavenumber Ek kαD. Another
∼
portant to notice that the fractal projector in front of important quantity is the spectrum averaged over the
the non-linear term in Eq. (3) is fundamental to ensure quenched disorder:
thatthenon-linearconvolutiondoesnotactivatethedec-
imated degrees of freedom during the system dynamics. E¯ = uˆ2 = 1 N(cid:88)maskθ(n) uˆ2 ,
It is interesting to note that the fractal decimation, as k (cid:104) k(cid:105) N k (cid:104) k(cid:105)
mask
n=1
well as any other Galerkin truncation, preserves the in-
viscid conservation of the first three moments of the ve- where with n = 1,...,N we indicate different real-
mask
locity field only. izations of the decimation mask. From these definitions,
3
t=70 t=70
t=50
t=50
) )
t t
x, x,
u( 0.2 t=30 v( 0.2 t=30
0.1 0.1
0 0
t=10 t=10
0.1 0.1
− (a) − (b)
0.2 0.2
− 0 1 2 3 4 5 6 − 0 1 2 3 4 5 6
x x
FIG. 1: (Color online) Representative plots of the solutions of the stochastically forced (a) Burgers equation and (b) the
decimatedBurgersequationforfractaldimensionD=0.99attimest=10,t=30,t=50,andt=70(respectivelyfromlower
black (black) line to upper red (gray) line). The velocities at different time are shifted upward on the y axes.
TABLEI: D,systemdimension;D=1denotestheordinarynon-decimatedBurgersequation[Eq. (2)],whileD<1represents
the decimated system as described in Eq. 3. N, number of collocation points. %(D), percentage of decimated wave numbers,
where the first value is related to the lower resolution used while the second value is related to the higher one. ν, value of the
kinematic viscosity. k , the range of forced wavenumbers. C , the mean energy injection, (cid:104)uf(cid:105). N , number of different
f f mask
random quenched masks. dt, time step used in the temporal evolution.
D N %(D) ν k C N dt
f f mask
1 216−218 0 8.0 10−5 [1:5−10] 0.01−0.05 0 5.5 10−5
0.99 216−219 8−10 2.5 10−5 [1:5−10] 0.01−0.05 32 2.3 10−5
0.97 216−218 23−27 9.0 10−6 [1:5−10] 0.01−0.05 64 2.0 10−5
0.95 216−218 36−40 5.0 10−6 [1:5−10] 0.01−0.05 64 1.7 10−5
0.90 216−218 59−64 2.0 10−6 [1:5−10] 0.01−0.05 64 1.6 10−5
0.80 216−218 83−87 8.0 10−7 [1:5−10] 0.01−0.05 96 1.5 10−5
0.70 216−218 93−95 6.5 10−7 [1:5−10] 0.01−0.05 96 1.5 10−5
one can infer the following relations: The existence of this quasi-singular behavior for the
spectral slope at D 1 might indicate the presence of
E¯ kD−1E kβD; β =α +D 1, (4) anintermediateasym∼ptoticsspoilingthetruebehaviorin
k k D D
(cid:39) ∼ −
the limit of vanishing viscosity. To confirm this possibil-
wherewehaveassumedthatthescalingpropertiesofthe ity,weshow,inFig. 2(b),theenergyspectrumE¯ versus
k
velocityfielddependonlyonthefractaldimensionDbut k for D =0.99 at various values of the viscosity and res-
are independent of the particular n-th realization of the olution. It is seen that there is a trend, when decreasing
decimation mask. To ensure the validity of Eq. (4) we viscosity, towards the development of an inertial range
need to use Nmask large enough to smooth out the gaps scaling E¯k k−3/2. This suggests that, asymptotically,
∼
produced by the different masks in the energy spectra. as ν 0, the true scaling exponent β 3/2 for any
D
→ →−
For the one-dimensional Burgers equation, because of D <1. Thisobservationwouldimplythatthecontinuous
the presence of the shock(s), the energy spectrum scales transitionseenintheinsetsofFig. 2(a)isaconsequence
as E¯ =E k−2 . of some intermediate asymptotics and that Fourier deci-
k k
InFig. 2(a∼)alog-logplotofE¯ versuskfor0.9 D mation is a singular perturbation for the spectral scaling
k
1.0 is shown. The mean energy spectrum E¯k k≤βD b≤e- properties.
∼
comes shallower when decreasing D. In the inset of Fig.
2(a), we show the dependency of the scaling exponents
onD. Itchangesfromβ = 2forD =1toβ = 3/2 B. Scaling in physical space
D D
for D (cid:46) 0.97, with a sharp t−ransition around D −0.97.
∼
In the same inset we also show the validity of the scaling In order to substantiate the relation between the
relation Eq.(4). change in the spectrum and the suppression of shocks
(a)
4
−−101101 ((aa)) DDDDDD======DD000000==......99999995095011 012012 νννν====1252....0505····11110000−−−−4566
22 kk−−33//22
¯¯EEkk−− 1.0 ¯¯EEkk −−11
loglog1010−−4343 −1.5 loglog1010 −−22 kk−−22
−− − 33
55 βD −−
−−−−66 −2.0.7αD 0.8 0.9 1.0 −−44 ((bb))
D 55
77 −−
−− 11 22 33 44 00 11 22 33 44
lloogg kk lloogg kk
1100 1100
FIG. 2: (Color online) (a) Mean energy spectrum, E¯ as a function of the wave numbers k; different lines and symbols
k
represent different fractal dimensions D (see the legend for details). Inset: the scaling exponents β (red line, downward
D
triangles)forthemeanenergyspectrumasafunctionofthedimensionD;exponentsα (blueline,upwardtriangles)obtained
D
fromthespectrumforasingleprojector. Thelatteriscomputedbyaveragingovertheexponentsobtainedfromeachindividual
projector. The black dashed line confirms the relation β +1−D = α as obtained in Eq. (4). The error bars for β are
D D D
obtained by halving the set of projectors used in the computation of the mean energy spectrum, while the error bars for α
D
are the standard deviation among all the values used in the calculation of the mean exponent. (b) Mean energy spectrum
(error bars are inside the symbols) at D = 0.99 for the following resolutions and values of the viscosity (see legend as well):
ν =1.0×10−4, N =217 (blackline); ν =2.5×10−5, N =218 (greensquaresandline); ν =5.0×10−6, N =218 (bluecircles
with line); and ν =2.5×10−6, N =219 (red triangles with line). The dashed black lines represent the scaling k−3/2 and k−2
as a guide to the eye.
1 5
−
S(2,D)(r) D=1 (b) 0
S(2,D)(r) 4 D 0.5
γ−
D)r()−2 2 D)r()3 D=D0.9=90.97 −1.
(2,S r)1.5 (4,F 0.7 0.8D0.9 1.0
log10−3 (2,D)ζ(0.51 log102 D=0.95
0 1 D=0.90
(a) 4 3 2 1 0
− − lo−g r − D=0.80
10
4 0
− 4 3 2 1 0 4 3 2 1 0
− − − − − − − −
log r log r
10 10
FIG. 3: (Color online) (a) S(2,D)(r) (red squares) and S(2,D)(r) (black solid line) measured at D = 0.80 and (inset) their
associated local slopes [Eq. (8)]. (b) The flatness, F(4,D)(r) versus r (log-log scale) for different dimensions D. The fit is done
in the range: 0.004≤r≤0.04 as illustrated by the two vertical dotted lines to yield the exponent γ as shown in the inset.
D
we analyze the physical space velocity field. This is also where δ v = v(x+r) v(x) and the angular brackets,
r
−
requiredtoaddresstheissueofanomalousscalingdueto , denote space and time averaging over the statis-
x,t
(cid:104)(cid:104)•(cid:105)(cid:105)
the lack of a clear definition of intermittency in Fourier tically stationary state. It is important to remark that
space. the spatial average is equivalent to an average over the
Intermittent features in turbulent flows are quantified quenched disorder. To prove this we notice that:
by the statistics of multi-points correlation function or
the so called structure functions:
S(p,D)(r)= δ vp ,
r x,t
(cid:104)(cid:104) (cid:105)(cid:105)
5
2 2
D=1 D=1
D=0.99 D=0.99
D=0.95 D=0.95
1.5 D=0.80 1.5 D=0.80
D) 1 D) 1
(p,ζabs (p,ζ
0.5 0.5
p/4 p/4
(a) (b)
0 0
0 1 2 3 4 5 6 0 1 2 3 4 5 6
p p
FIG. 4: (Color online) Inertial range scaling exponents (a) ζ(p,D) and (b) ζ(p,D) for the structure functions (a) with and (b)
abs
without the use of absolute values versus the order p; the different symbols are related to different dimensions D (see legend),
the dashed lines are the bi-fractal behavior of the 1D Burgers equation. The values of the exponents are estimated as the
best fit of local scaling exponents and the error bars are estimated from the variations of the local scaling exponents within
thefittingrange. Wenotethatζ(3,D) doesnotsatisfythe1D Ka´rma´n-Howarthanalyticalrelationbecauseofthecompetition
abs
between the leading and sub-leading terms introduced by the decimation in the scaling of the velocity field; in contrast the
Karman-Howarth analytical relation is satisfied for the case without the absolute values (see text).
S(2,D)(r)=(cid:90) dx(v¯(x+r) v¯(x))2 =(cid:90) dk(eikr 1)E¯ = 1 N(cid:88)mask(cid:90) dk(eikr 1)θ(n)E =S(2,D)(r), (5)
− − k N − k k
mask
n=1
where we have used the scaling properties of the prob- press intermittency in hydrodynamics (as has also been
ability defining the decimation mask and the fact that seen in Ref. [20]). We cannot refrain from noticing that
dkθ(n) lawdkkD−1, where the symbol law stands for sta- this seems to be in contrast with the usual phenomenol-
tistikcally∼“in law”, i.e., the two sides ha∼ve the same scal- ogy of cascade dynamics, built in terms of local-Fourier
ing properties when averaged on different realizations of interactions.
the fractal projector. This relation is validated in Fig.
3(a)wherethesecond-orderstructurefunctionsobtained
bothfromasinglemask(continuousblackline)andfrom IV. HIGHER ORDER STATISTICS
an average over different realizations of the quenched
masks (square symbols in red) are shown to be identi- The results obtained in the previous section lead us to
cal. For this reason henceforth, we stop distinguishing address the question of whether intermittency is indeed
between S¯p(r) and Sp(r). To understand the effects of washed out by any small perturbation of the Fourier dy-
decimation on intermittency, we measure the flatness of namics – bad news for modeling – or if it is masked by
structure functions: newleadingfluctuationsintroducedbythemodifiednon-
linear dynamics. To answer this question, we perform a
systematic analysis of the scaling properties of structure
S(4,D)(r)
F(4,D)(r)= rγD (6) functions by changing the fractal Fourier dimension D.
[S(2,D)(r)]2 ∼ It is important to decompose the structure function into
contributions from the negative and positive increments
as a function of r for different values of D. Let us stress
of the velocity field [17]. We thus define
thatforthe1DBurgersequations,phenomenologicaland
theoretical arguments [17] predict ζ(p) = min(p,1) (see
S(p,D)(r)= (δ+v)p ; S(p,D)(r)= (δ−v)p ,
Fig. 2), which give for the flatness the scaling r−1, [Fig. + (cid:104)(cid:104) r (cid:105)(cid:105)x,t − (cid:104)(cid:104) r (cid:105)(cid:105)x,t
3(b), inset]. As shown in Fig. 3(b) we find that scal-
where δ+v δ v > 0 and δ−v δ v < 0,
ing exponents, γD, present the same sharp transition whence trhe st≡ructurre function S(p,Dr)(r)≡= Sr(p,D)(r) +
already observed in the slope of the energy spectra for +
1 D 0.97 [Fig. 2(a), inset]. Thus, surgeries on the ( 1)pS(p,D)(r). To improve the statistics, odd-order
≤ ≤ − −
Fourier space and dimensional reduction seem to sup- structure functions are often measured in terms of the
6
absolutevalueofvelocityincrements; inthiscasewewill of intermittency in Burgers flows under fractal Fourier
obviously have S(p,D)(r) = S(p,D)(r) + S(p,D)(r). To decimation, similar to what has been observed for the
abs + −
study the scaling properties it is customary to analyze Navier-Stokes evolution in Ref. [20]. The deviation from
logarithmic local slopes: thevaluesp/4fortheodd-ordermomentsisexplainedby
noticingthatthethird-orderstructurefunctionmustsat-
isfyananalyticalrelationsimilartotheK´arm´an-Howarth
dlogS(p,D)(r) 4/5 law of Navier-Stokes, namely: S(3,D)(r) = 6(cid:15)r for
ζ(p,D)(r) = abs ; (7) −
abs dlogr allDandwhere(cid:15)isthemeanenergydissipation. Indeed,
Fig. 4(b)clearlysupportsthisstatement. Apossibleway
dlogS(p,D)(r)
ζ(p,D)(r) = . (8) torationalizetheseapparentlycontradictoryresultsisto
dlogr
supposethatdecimationintroducesadistributednoiseat
all scales, leading to a typical scaling δ v r1/4 on top
The scaling exponents of order p in the inertial range r ∝
ofanunderlyingBurgers-likedynamics. Ifthisistrue, it
are obtained as a best fit to the local exponents in the
should be detectable by looking separately at positive or
interval of scales where they are close to a constant. In
negative velocity increments. As a result, we suggest the
Fig. 4 we show the result for both ζ(p,D) and ζ(p,D) (see
abs presence of two different asymptotics:
figure captions for details).
Fromacomparisonofthetwofiguresonecanconclude (cid:40)S(p,D)(r)=rp/4+smooth;
a few important facts. First, there is a clear tendency + (9)
foreven-ordermomentstoapproachthenon-intermittent S−(p,D)(r)=rp/4+r+smooth,
scaling behavior with exponent p/4 as soon as D < 1;
the agreement being almost perfect already at D 0.95. where the first term on the right hand side of the equa-
Second, the odd-order moments of the structure≤func- tions should have pre-factors that go to zero for D 1.
→
tions, defined without absolute values, seem to maintain InEq. (9)theBurgersscaling r ispresentonlyforthe
∝
a memory of the original non-decimated Burgers behav- negative increments and smooth denotes the sub-leading
ior, namely ζ(p,D)=1 p 1, even for small fractal di- differentiable terms induced by the viscous contribution
mensionsD 1. Letus∀no≥tethatthisbehaviorisnoten- rp.
tirelyunexpe(cid:28)cted. First,theemergenceofalinearscaling ∝In order to quantitatively check the above prediction,
(p,D)
p/4 is in agreement with the observation of the spectral weperformaseriesofsystematicfitstoS (r)byusing
±
slope β = 3/2 and with the absence of intermittency. the following interpolation expression for the asymptotic
D
−
Figure 4(a) is thus a demonstration of the suppression behavior (9):
 (cid:18) (cid:19)
S+(p,D)(r) = (r2+Aη(+2p),(Dp−)rpp/4)/2 +B+(p,D)rp (cid:0)1+ Lr(cid:1)c+;
(cid:18) (cid:19) (10)
S−(p,D)(r) = (r2+Aη(−2p),(Dp−)rpp/4)/2 + (r2B+−η(p2,)D(p)−rp1)/2 (cid:0)1+ Lr(cid:1)c−,
where A(p,D),B(p,D) are fitting constants and η is the case of negative increments, we also show the best fit
± ±
dissipative scale such that for r < η, S(p,D)(r) rp. withandwithouttheshockcontributiontohighlightthe
The overall factor (1 + r/L)c± is used to saturat∝e the importanceoftheshocktoreconstructtherightbehavior
inertial range scaling for r beyond the forcing length for p=2 in Fig. 6(a) and for p=3 in Fig. 6(b). As one
scale. L and c are estimated as the best-fitting pa- can see, the Eqs. (10) are able to reconstruct the local
±
rameters. From Eq. (10) it is easy to recognize that scalingpropertiesinarobustway,showingthatourphe-
the first term on the right hand side of both equations nomenological model is not incompatible with the data.
representstheeffectintroducedbythedecimation, while Finally we measure the probability density function of
the second term (present only for negative increments) the velocity increments at different scales (Fig. 7). This
represents the standard shock-dominated Burgers scal- clearly shows the emergence of non-trivial fluctuations
ing (plus viscous contributions). Clearly, if A A for positive velocity increments at decreasing fractal di-
+ −
for all D, we have moments of even order that are∼dom- mension D; such fluctuations are almost absent in the
inated by the p/4 scaling while moments of odd order standard one-dimensional Burgers case. In summary, all
have the usual Burgers scaling. In Figs. 5(a) and 5(b) our results indicate that while on the one hand the nu-
we show the best fit by using these expressions for the merical evidence point toward a robustness of the shock
third-order moment (p = 3) and D = 0.95,0.8 for both structure (as also visually confirmed from Fig. 1), on
the (a) negative and (b) positive increments. For the the other hand, decimation introduces important fluctu-
7
4 D=0.95 4 D=0.95
− D=0.80 − D=0.80
D)r() −6 3 D)r() −6 3
(3,S− r)2 (3,S+ r)2
log10 −8 (3,D)ζ(−1 log10 −8 (3,D)ζ(+1
10 (a) 0 log10r 10 (b) 0 log10r
− 4 2 0 − 4 2 0
− − − −
4 3 2 1 0 4 3 2 1 0
− − − − − − − −
log r log r
10 10
FIG. 5: (Color online) (a) Structure functions for D = 0.80 (blue stars) and D = 0.95 (red circles) for (a) the negative
increments S(3,D)(r) and (b) positive increments structure functions S(3,D)(r); the black solid lines are the respective fitting
− +
functions [Eq. (10)]. In the insets, with the same legend, we show the associated local slopes.
2 3
2.5
1.5
(a)
2 (b)
) )
r r
D)( 1 D)( 1.5
(2,ζ− (3,ζ−
1
0.5
Data Data
Fitwithoutshock 0.5 Fitwithoutshock
Fitwithshock Fitwithshock
0 0
4 3 2 1 0 4 3 2 1 0
− − − − − − − −
log r log r
10 10
FIG.6: (Coloronline)(a)LocalslopesofthestructurefunctionsassociatedwiththenegativeincrementsatD=0.95for(a)
thesecondorder[S(2,D)(r)]and(b)thethirdorder[S(3,D)(r)];theblacksolidlinesareobtainedfromthefittingfunction[Eq.
− −
(10)] and the dashed lines from the fit obtained without the shock contribution by setting B(p,D) =0 in [Eq. (10)].
+,−
ations that spoil the scaling of the original undecimated the fractal decimation. The fractal mask can be seen as
equation without modifying the existence of a constant- an extra, ad-hoc removal of non-linear couplings at all
flux of energy from large to small-scales. scalesand,assuch,asortofpower-lawexternal“energy-
conserving” noise. It is not unphysical to suppose that
due to the power-law dependence on the wavenumber,
a different weight between local and non-local interac-
V. CONCLUSIONS
tions in Fourier space is introduced, making the latter
more important then in the usual D = 1 case. Given
Let us now turn to a few theoretical considerations
allthis,itisconceivablethatanextradecorrelationtime
for understanding the behavior of ζ(p,D). We first recall
of the order of τ (k) 1/k appears, leading to a slow
dec
that just like any Galerkin truncation [22], the fractal ∝
downoftheenergytransfermechanisms,asisthecasefor
Fourier decimation constrains the number of conserved
Alfv´en waves in MHD [23] or in the presence of a rapid
quantities to the first three moments of the velocity in
distortionmechanism[24]. Typically, thisleadstoanes-
the Burgers equation. In particular, this allows the con-
timate for the energy flux (cid:15) = kE(k)/τ (k), where the
tr
servation of a cubic moment whose relative flux would
transfer time is given in terms of a golden mean between
yield ζ = 1, which, in turn, would be consistent with
4 the eddy-turn-over time τ (k) and the decorrelation
eddy
our numerical result ζp = p/4. Another possible expla- time τ (k), τ (k) = τ2 /τ (k). If this is the case,
nation for the E(k) k−3/2 scaling is the idea that a dec tr eddy dec
new decorrelation me∝chanism in the shell-to-shell energy considering that τeddy ∝ (k3E(k))−1/2, we arrive at the
estimate E(k) k−3/2.
transfer across Fourier modes might be introduced by ∝
8
getherwiththeonesshownheresuggestthatthebuildup
1
D=1 of small-scale intermittent fluctuations in physical space
D=0.99
(shocks) is indeed the outcome of an entangled temporal
0 D=0.95
correlations amongst many (all?) Fourier modes. An-
D=0.80
otherinterestingpotentiallyusefulmethodologyistoap-
1
F − ply proper orthogonal decomposition of Fourier ampli-
D
tudes and phases correlations [27].
P 2
0 − In this paper, we have presented a set of theoretical
1
g
o and numerical results concerning the evolution of the
l 3
− one-dimensional stochastically forced Burgers equation
decimated on a fractal Fourier set. Decimation leads,
4
− very quickly, to a suppression of the shock-dominated
statistics, indicating that the bifractal scaling properties
5
− -15 -10 -5 0 5 of the original equation are very sensitive to the details
of the dynamical evolution. Similar results have also
δ v/ (δ v)2 1/2
r r
h i been recently obtained for the more complicated case of
the dynamics of fully developed incompressible turbu-
FIG. 7: (Color online) The probability density function
lent flows in three dimensional Navier-Stokes equations.
(PDF) of the velocity increments at a scale r ∼ 0.005; the
Some properties connected to the existence of shock-like
PDF is normalized by its standard deviation. The different
lines correspond to different dimensions D as shown in the solutions are nevertheless robust, but sub-leading. Our
legend. resultsindicatethattheexistenceofstronglocalizedfluc-
tuations in Burgers is the result of highly entangled cor-
relationsamongallFouriermodes. Thismightbeimpor-
tant to develop models for the nonlinear evolution based
Let us notice that other decimation protocols might
on suitable reduction (and replacement) of a subset of
be imagined. In particular, one can consider perform-
the original degrees of freedom.
ing a selective decimation of a single class of triads (e.g.,
local or nonlocal), in order to probe the main mecha-
nisms leading to the formation of small-scales shocks in
the dynamical evolution. In this case decimation cannot
Acknowledgments
be univocally defined in terms of each wavenumber, i.e.
onewavenumbermightbelongtoalocalornon-localtri-
ads depending on the other two. Hence a selected triads MB and LB acknowledge funding from the Euro-
reductioncanbedoneonlyinsidethenonlinearconvolu- peanResearchCouncilundertheEuropeanUnion’sSev-
tionterm,accessibleviaafullyspectralcodewithstrong enthFrameworkProgramme, ERCGrantAgreementNo
limitationinthenumericalresolutionachievable;seeRef. 339032. LB and SSR thank COST ACTION MP1305
[25]. for support. SSR acknowledges the support of the Indo-
Moreover,arecentstudy[26]hasshownthathighlynon- French Center for Applied Mathematics (IFCAM) and
trivial time correlations among Fourier triads are con- AIRBUS Group Corporate Foundation Chair in Mathe-
nectedtothepresenceofintermittencyinphysicalspace. matics of Complex Systems established in ICTS and the
It is not obvious a priori that reducing Fourier inter- hospitality of the Department of Physics, University of
action will lead to a time de-synchronization of the en- Rome “Tor Vergata”, Rome, Italy and the Observatoire
ergy exchange among triads. The results of Ref. [26] to- de la Cˆote d’Azur, Nice, France.
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