a r X i v : n l i n / 0 4 0 6 0 4 9 v 2 [ n l i n . C D ] 1 8 F e b 2 0 0 5
Is Multiscaling an Artifact in the Stochastically Forced Burgers Equation?
Dhrubaditya Mitra,
1,2
J´er´emie Bec,
2,3
Rahul Pandit,
1,
∗
and Uriel Frisch
2
1
Centre for Condensed Matter Theory, Department of Physics,Indian Institute of Science, Bangalore 560012, India
2
D´epartement Cassiop´ee, Observatoire de la Cˆ ote d’Azur, BP4229, 06304 Nice Cedex 4, France
3
Dipartimento di Fisica, Universit`a La Sapienza, P.zzle Aldo Moro 2, 00185 Roma, Italy
(Dated: February 8, 2008)We study turbulence in the onedimensional Burgers equation with a whiteintime, Gaussianrandom force that has a Fourierspace spectrum
∼
1
/k
, where
k
is the wave number. From veryhighresolution numerical simulations, in the limit of vanishing viscosity, we ﬁnd evidence for multiscalingof velocity structure functions which cannot be falsiﬁed by standard tests. We ﬁnd a new artifact inwhich logarithmic corrections can appear disguised as anomalous scaling and conclude that bifractalscaling is likely.
PACS numbers: 47.27 Gs, 05.45a, 05.40a
Homogeneous, isotropic ﬂuid turbulence is often characterized by the order
p
velocity structure functions
S
p
(
ℓ
) =
[
{
v
(
x
+
ℓ
)
−
v
(
x
)
} ·
(
ℓℓ
)]
p
, where
v
(
x
) is thevelocity at the point
x
and the angular brackets denote an average over the statistical steady state of theturbulent ﬂuid. For separations
ℓ
in the inertial range,
η
d
≪
ℓ
≪
L
, one has
S
p
(
ℓ
)
∼
ℓ
ζ
p
. Here
η
d
is the smalllength scale at which dissipation becomes important;
L
isthe large length scale at which energy is fed into the ﬂuid.The 1941 theory (K41) of Kolmogorov [1] predicts
simple scaling
with exponents
ζ
K
41
p
=
p/
3. By contrast, experiments and direct numerical simulations (DNS) suggest
multiscaling
with
ζ
p
a nonlinear, monotonically increasing, convex function of
p
, not predictable by dimensionalanalysis [2]. However, the Reynolds numbers achieved inDNS are limited, so the exponents
ζ
p
have to be extractedfrom numerical ﬁts over inertial ranges that extend, atbest, over a decade in
ℓ
. The processing of experimentaldata – although they can achieve much higher Reynoldsnumbers – involves other wellknown diﬃculties [3]. It isimportant therefore to establish, or rule out, multiscaling of structure functions in simpler forms of turbulence,such as passivescalar, passivevector or Burgers turbulence. Signiﬁcant progress, both analytical and numerical, has been made in conﬁrming multiscaling in passivescalar and passivevector problems (see, e.g., Ref. [4] fora review). The linearity of the passivescalar and passivevector equations is a crucial ingredient of these studies,so it is not clear how they can be generalized to ﬂuidturbulence and the Navier–Stokes equation.Here we revisit the onedimensional, Burgers equation with stochastic selfsimilar forcing, studied earlierin Refs. [5, 6]. It is by far the simplest
nonlinear
partialdiﬀerential equation (PDE) that has the potential to display multiscaling of velocity structure functions [6]; andit is akin to the Navier–Stokes equation. In particular,we investigate the statistical properties of the solutionsto
∂
t
u
+
u∂
x
u
=
ν∂
xx
u
+
f
(
x,t
)
,
(1)in the limit of vanishing viscosity
ν
→
0. Here
u
is thevelocity, and
f
(
x,t
) is a zeromean, spaceperiodic Gaussian random force with
ˆ
f
(
k
1
,t
1
) ˆ
f
(
k
2
,t
2
)
= 2
D
0

k

β
δ
(
t
1
−
t
2
)
δ
(
k
1
+
k
2
) (2)and ˆ
f
(
k,t
) the spatial Fourier transform of
f
(
x,t
). We restrict ourselves to the case
β
=
−
1 and assume spatial periodicity of period
L
. Earlier studies [5, 6] suggested that
Eqs. (1) and (2), with
β
=
−
1, show a nonequilibriumstatistical steady state with
bifractal scaling
: this meansthat velocity structure functions of order
p
≤
3 exhibitselfsimilar scaling with exponents
p/
3 and implies a K41type
−
5
/
3 energy spectrum, predictable by dimensionalanalysis, whereas those of order
p
≥
3 have exponents allequal to unity being dominated by the ﬁnite number of shocks, with
O
(
L
1
/
3
) strength, typically present in theperiodic domain; this bifractal scaling is somewhat similar to that observed when the Burgers equation is forcedonly at large spatial scales [7, 8].
We overcome the limitations of these earlier studies [5, 6] by adapting the algorithm of Refs. [9, 10] to
develop a stateoftheart technique for the numerical solution of Eqs. (1) and (2), in the
ν
→
0 limit. This yieldsvelocity proﬁles (Fig. 1 a) with shocks at all length scalesresolved. Structure functions [Figs. (1 b) and (1 c)] ex
hibit powerlaw behavior over nearly three decades of
r
;this is more than two decades better than in Ref. [5]. Inprinciple it should then be possible to measure the scalingexponents [Figs. (1 b)] with enough accuracy to decidebetween bifractality and multiscaling. A naive analysis[Fig. (2 a)] does suggest multiscaling [15]. However, given
that simple scaling or bifractal scaling can sometimes bemistaken for multiscaling in a variety of models [12, 13], it
behooves us to check if this is the case here. We describebelow our numerical procedure and the various tests wehave carried out.
2
0 2 4 6−6−4−20246x
u ( x ) o r f ( x )
(a)
−6 −4 −2 0−3−2−1012log
10
(r/L)
l o g
1 0
[ S
a b s 3
( r ) ]
−4 −20.60.81(b)
−6 −4 −2 0−3−2−10123log
10
(r/L)
l o g
1 0
[ S
a b s 4
( r ) ]
−4 −20.60.81(c)
FIG. 1: (a) Representative snapshots of the force
f
and the velocity
u
(jagged line), in the statistically stationary r´egime; thevelocity develops smallscale ﬂuctuations much stronger than those present in the force. Loglog plots of the structure function
S
abs
p
(
r
) versus
r
for
N
= 2
20
and (b)
p
= 3 and (c)
p
= 4. The straight line indicates the leastsquares ﬁt to the range of scaleslimited by the two vertical dashed lines in the plots. The resulting multiscaling exponents
ξ
p
(see text) are shown by horizontallines in the insets with plots of the local slopes versus
r
.
In our simulations we use
L
= 2
π
and
D
0
= 1 withoutloss of generality. The spatial mesh size is
δx
=
L/N
where
N
is the number of grid points. To approximatethe forcing, we use the “kicking” strategy of Ref. [9]in which the whiteintime force is approached by shotnoise. Between successive kicks we evolve the velocityby using the following wellknown result on the solutionsto the unforced Burgers equation in the limit of vanishing viscosity (see, e.g., Ref. [8]): the velocity potential
ψ
(such that
u
=
−
∂
x
ψ
) obeys the maximum principle
ψ
(
x,t
′
) = max
y
ψ
(
y,t
)
−
(
x
−
y
)
2
2(
t
′
−
t
)
;
t
′
> t.
(3)The search for the maxima in Eq. (3) requires only
O
(
N
log
2
N
) operations [10] because, under Burgers dynamics, colliding Lagrangian particles form shocks anddo not cross each other. At small scales we want to haveunforced Burgers dynamics with wellidentiﬁable shocks.At least four mesh points are needed for unambiguousidentiﬁcation of a shock; since the maximum wavenumber is
N/
2, we set ˆ
f
(
k,t
) = 0 beyond an ultravioletcutoﬀ Λ =
N/
8.Speciﬁcally, at time
t
n
=
nδt
we add
f
n
(
x
)
√
δt
tothe Burgers velocity
u
(
x,t
), where the
f
n
(
x
)s are independent Gaussian random functions with zero meanand a Fourierspace spectrum
∼
1
/k
, for
k <
Λ. Thetime step
δt
is chosen to satisfy the following conditions(
δx/
2
u
0
)
< δt
≃
(1
/L
Λ)
2
/
3
(
L/u
0
), where
u
o
is the characteristic velocity diﬀerence at large length scales
O
(
L
).The ﬁrst ensures that a typical Lagrangianparticle movesat least half the meshsize in time
δt
(otherwise, it wouldstay put)[16]; the second, which expresses that
δt
is comparable to the turnover times at the scale
L/
Λ, guarantees that, at scales larger than
L/
Λ the timestep,
δt
,is small compared to all dynamically signiﬁcant times,but still permits the formation of individual shocks atsmaller scales. Finally, as our simulations are very long,for the stochastic force we use a goodquality random
Run
N δt
Λ
τ
L
T
tr
T
av
B1 2
20
5
×
10
−
4
2
17
1
.
0 2
.
0 22B2 2
18
1
×
10
−
4
2
15
1
.
0 2
.
0 20B3 2
16
1
×
10
−
4
2
13
1
.
0 2
.
0 120TABLE I: Diﬀerent parameters used in our runs B1, B2 andB3.
τ
L
≡
L/u
0
is the equivalent of the largeeddyturnovertime. Data from
T
tr
time steps are discarded so that transients die down. We then average our data over a time
T
av
.
number generator with a long repeat period of 2
70
dueto Knuth [11]. The main characterisitics of the runs performed are summarized in Table I.In addition to the usual structure functions, we haveused
S
abs
p
(
r
), deﬁned by
S
abs
p
(
r
)
≡ 
δu
(
x,r
)

p
∼
r
ξ
p
,
(4)
δu
(
x,r
)
≡
u
(
x
+
r
)
−
u
(
x
)
,
(5)from which we extract the exponents
ξ
p
. For each valueof
N
we have calculated
ξ
p
for
p
=
m/
4, with integers1
≤
m
≤
20. Figure (2 a) summarizes the results of our calculations concerning the exponents
ξ
p
, for
N
=2
16
,
2
18
, and 2
20
; any systematic change in the values of these exponents with
N
is much less than the error barsdetermined by the procedure described below. Thus inall other plots we present data from our simulations with
N
= 2
20
grid points. The representative loglog plots of Figs. (1 b) and (1 c) of
S
abs
p
(
r
) for
p
= 3 and 4 showpowerlaw r´egimes that extend over nearly three decadesof
r/L
. We obtain our estimates for the exponents
ξ
p
asfollows: for a given value of
p
we ﬁrst determine the localslopes of the plot of log
S
abs
p
versus log
r
by leastsquaresﬁts to all triplets of consecutive points deep inside thepowerlaw r´egime [17]. These regions extend over oneand half decade of
r/L
as shown in Figs. (1 b) and (1 c).
The value of
ξ
p
we quote [Fig. (2 a)] is the mean of these
3local slopes; and the error bars shown are the maximumand minimum local slopes in these regions.Figure (2 a) shows that our results for
ξ
p
, indicatedby circles for
N
= 2
20
, deviate signiﬁcantly from thebifractalscaling prediction (full lines). As we shall seethis deviation need not necessarily imply “multiscaling”for the structure functions. We have also considered thepossibility of already wellunderstood artifacts, such asthe role of temporal transients [12] and ﬁnitesize eﬀectswhich can round sharp bifractal transitions [13], and decided that they do not play any major role in the presentproblem [18].Consider
P
c
(
s
), the cumulative probability distribution function of shock strengths
s
. Simple scaling arguments predict
P
c
(
s
)
∼
s
γ
, with
γ
=
−
3, which followsby demanding that
P
c
(
s
) remain invariant if lengths arescaled by a factor
λ
and velocities by
λ
−
1
/
3
. One of the signatures of multiscaling would be deviations of
γ
from this scaling value. However, the following argument favors
γ
=
−
3 : the total input energy,
E
in
=
Λ
k
0
D
(
k
)
dk
∼
lnΛ, where Λ is the ultraviolet cutoﬀ.In the limit of vanishing viscosity, the energy dissipation in the Burgers equation occurs only at the shocksand is proportional to the cube of the shock strength [8],so the total energy dissipation is Ω
∼
s
max
s
min
P
(
s
)
s
3
ds
.Here
P
(
s
) = (
dP
c
(
s
)
/ds
)
∼
s
γ
−
1
is the probability distribution function (PDF) of shock strengths and
s
min
and
s
max
are, respectively, the minimum and maximum shockstrengths. A steady state can occur only if
E
in
and Ωhave the same asymptotic properties as Λ
→ ∞
, i.e.,Ω
∼
ln(Λ); this requires
γ
=
−
3. By contrast we ﬁnd
γ
≃ −
2
.
7 [Fig. (2 b)] by a naive leastsquares ﬁt to thetail of
P
c
(
s
) [19] . This suggests that the results of oursimulation are far from the limit Λ
→∞
, although morethan a million grid points are used. Hence the “anomalous” exponents in Fig. (2 a) might well be suspect.To explore this further, consider the thirdorder structure function of velocity diﬀerences, without the absolutevalue, namely,
S
3
(
r
)
≡
δu
3
. From Eqs. (1) and (2) fol
lows the exact relation16
S
3
(
r
) =
r
0
F
(
y
)
dy,
(6)where
F
(
y
) is the spatial part of the force correlationfunction, deﬁned by
f
(
x
+
y,t
′
)
f
(
x,t
)
=
F
(
y
)
δ
(
t
−
t
′
).We obtain this analog of the von K´arm´an–Howarth relation in ﬂuid turbulence by a simple generalization of theproof given in Ref. [9] for the Burgers equation forced deterministically at large spatial scales. An explicit checkof Eq. (6) provides a stringent test of our simulations[Fig. (2 c) inset]. Furthermore, Eq. (6) implies that
S
3
(
r
)
∼
r
log(
r
) for small
r
and thus should displaysigniﬁcant curvature in a loglog plot, as is indeed seenin Fig. (2 c). By contrast
S
abs
p
(
r
) [Fig.(2 c)] displaysmuch less curvature and can be ﬁtted over nearly threedecades in (
r/L
) to a power law with an “anomalous”exponent of 0
.
85. This anomalous behavior is probably an artifact as we now show. Let us deﬁne thepositive (resp., negative) part of the velocity increment
δ
+
u
(resp.,
δ
−
u
) equal to
δu
when
δu
≥
0 and to zerowhen
δu <
0 (resp., to
δu
when
δu
≤
0 and to zerowhen
δu >
0). Obviously
S
3
(
r
) =
(
δ
+
u
)
3
+
(
δ
−
u
)
3
,whereas
S
abs
p
(
r
) =
(
δ
+
u
)
3
−
(
δ
−
u
)
3
. The loglog plotof
(
δ
+
u
)
3
= (1
/
2)[
S
3
(
r
)+
S
abs3
(
r
)] in Fig. (2 c) is muchstraighter than those for
S
3
(
r
) and
S
abs3
(
r
) and leads toa scaling exponent 1
.
07
±
0
.
02, very close to unity. Fora moment assume that
(
δ
+
u
)
3
indeed has a scaling exponent of unity. Given that
S
3
(
r
) has, undoubtedly, alogarithmic correction, it follows that
S
abs3
(
r
) has (exceptfor a change in sign) the same logarithmic correction inits leading term (for small
r
) but
diﬀers by a subleading correction proportional to
r
. This subleading correctionis equivalent to replacing
r
log(
r
) by
r
log(
λr
) for a suitably chosen factor
λ
. In a loglog plot this shifts thegraph away from where it is most curved and thus makesit straighter, albeit with a (local) slope which is not unity.An independent check of
(
δ
+
u
)
3
∼
r
is obtainedby plotting the cumulative probabilities, Φ
c
, of positive and negative velocity increments (for a separation
r
= 800
δx
) in Fig. (2 b) [20]. For positive increments
Φ
c
falls oﬀ faster than any negative power of
δu
, but,for negative ones, there is a range of increments overwhich Φ
c
∼ 
δu

−
3
, the same
−
3 law seen in
P
c
(
s
) earlier. Indeed the negative increments are dominated bythe contribution from shocks. Just as
P
c
(
s
) has cutoﬀs
s
min
and
s
max
, Φ
c
has cutoﬀs
u
−
min
(
r
) and
u
−
max
(
r
)for negative velocity increments. Since Φ
c
falls oﬀ as

δu

−
3
,
u
−
max
can be taken to be
∞
; furthermore as thePDF of velocity diﬀerences, Φ(
δu
)
≡
d
(Φ
c
)
/d
(
δu
), mustbe normalizable, we ﬁnd
u
−
min
(
r
)
∼
r
1
/
3
. We now knowenough about the form of Φ to obtain, in agreement withour arguments above, that
S
3
(
r
)
≈ −
Ar
ln(
r
) +
Br
and
S
abs3
(
r
)
≈
Ar
ln(
r
) +
Br
, whence
(
δ
+
u
)
3
≈
Br
. Thepresence of this cutoﬀ yields a logarithmic term in both
S
3
and
S
abs3
but with diﬀerent sign agreeing with thearguments given in the previous paragraph.By a similar approach, we ﬁnd
S
4
(
r
)
≈
Cr
−
Dr
4
/
3
,where
C
and
D
are two positive constants. The negativesign before the subleading term (
r
4
/
3
) is crucial. It implies that, for any ﬁnite
r
, a naive powerlaw ﬁt to
S
4
canyield a scaling exponent less than unity. The presence of subleading, powerlaw terms with opposite signs also explains the small apparent “anomalous” scaling behaviorobserved for other values of
p
in our simulations. A similar artifact involving two competing powerlaws has beendescribed in Ref. [14].In conclusion, we have performed veryhighresolutionnumerical simulations of the stochastically forced Burgers equation with a 1
/k
forcing spectrum. A naive interpretation of our data shows apparent multiscaling phenomenon. But our detailed analysis has identiﬁed ahithertounknown numerical artifact by which simple bis
4
0 1 2 3 4 500.20.40.60.81p
ξ
p
0 1 2 3 4 5−0.100.2(a)
−6 −4 −2 0−7−6−5−4−3−2−10log
10
(s)
l o g
1 0
[ P
c
( s ) ]
−2 −1 0 1−6−4−20(b) log
10
[
δ
u(r)]
l o g
1 0
[
Φ
c
(
δ
u ) ]
−5 −4 −3 −2 −1−3−2−1012
−6 −4 −2−3−2−10
log
10
(r)
l o g
1 0
[ s t r u c . f u n c . ]
( 1 / 6 ) l o g
1 0
[ S
3
( r ) ]
log
10
(r)
(c)
FIG. 2: (a) The multiscaling exponents
ξ
p
versus order
p
for Eqs. (1) and (2) with
N
= 2
16
(
⋄
)
,
2
18
(
∗
), and 2
20
(
◦
) grid points.Error bars (see text) are shown for the case
N
= 2
20
. The deviation of
ξ
p
from the exponents for bifractal scaling (full lines),shown as an inset, suggest naive multiscaling. (b) Loglog plots of the cumulative probability distribution function
P
c
(
s
) versusshock strengths
s
obtained from an average over 1000 snapshots. A leastsquares ﬁt to the form
P
c
(
s
)
∼
s
γ
, for the dark pointsin the range
−
5
log
10
[
P
(
s
)]
−
2
.
5, yields
γ
=
−
2
.
70; the simplescaling prediction
γ
=
−
3 is indicated by the straightline. The inset shows loglog plots of the cumulative probability distribution function, Φ
c
[
δu
(
r
)], (dashed line : positive
δu
,continuous line : negative
δu
) versus the velocity diﬀerence
δu
(
r
) for length scale
r
= 800
δx
. (c) Loglog plots of
S
3
(
r
) (crosses),
S
abs3
(
r
) (dashed line) and
(
δ
+
u
)
3
(squares) versus
r
. The continuous line is a leastsquare ﬁt to the range of points limited bytwo vertical dashed lines in the plot. Inset: An explicit check of Eq. (6) from our simulations, plotted on a loglog scale. Thedashed line is the righthand side of Eq. (6); the lefthand side of this equation has been obtained for
N
= 2
20
(
◦
) (run B1).
caling can masquerade as multiscaling. Our work illustrates that the elucidation of multiscaling in spatiallyextended nonlinear systems, including the Navier–Stokesequation, requires considerable theoretical insight thatmust supplement stateoftheart numerical simulationsand experiments.We thank E. Aurell, L. Biferale, A. Lanotte, and V.Yakhot for discussions. This research was supported bythe IndoFrench Centre for the Promotion of AdvancedResearch (Project 24042), by CSIR (India), and by theEuropean Union under contracts HPRNCT200000162and HPRNCT200200300. Additional computationalresources were provided by CHEP (IISc).
∗
Also at Jawaharlal Nehru Centre For Advanced ScientiﬁcResearch, Jakkur, Bangalore, India[1] A.N. Kolmogorov,
Dokl. Acad. Nauk USSR
30
, 9 (1941).[2] U. Frisch,
Turbulence: The Legacy of A.N. Kolmogorov
(Cambridge University Press, Cambridge, 1996).[3] F. Anselmet, Y. Gagne, E.J. Hopﬁnger and R.A. AntoniaJ. Fluid. Mech.
140
, 63 (1984) and references therein.[4] G. Falkovich, K. Gaw¸edzki, and M. Vergassola, Rev.Mod. Phys.
73
, 913 (2001) and references therein.[5] A. Chekhlov and V. Yakhot, Phys. Rev. E
52
, 5681(1995).[6] F. Hayot and C. Jayaprakash, Phys. Rev. E
56
, 4259(1997);
ibid
,
56
, 227 (1997).[7] W. E, K. Khanin, A. Mazel and Ya. Sinai, Phys. Rev.Lett.
78
, 1904 (1997).[8] U. Frisch and J. Bec in
New Trends in Turbulence
, A. Yaglom, F. David, and M. Lesieur eds., Les Houches SessionLXXIV (Springer EDPSciences, 2001) p 341.[9] J. Bec, U. Frisch, and K. Khanin, J. Fluid Mech.
416
,239 (2000).[10] A. Noullez and M. Vergassola, J. Sci. Comp.
9
, 259(1994).[11] D.E. Knuth,
The Art of Computer Programming
, Vol.2 (Addison Wesley Longman, Singapore, 1999) pp. 184187. P. Grassberger, New J. Phys.
4
, 17 (2002).[12] A. Kundagrami and C. Dasgupta, Physica A,
270
, 135(1999).[13] E. Aurell, U. Frisch, A. Noullez and M. Blank, J. Stat.Phys.,
88
, 1151 (1997).[14] L. Biferale, M. Cencini, A.S. Lanotte, M. Sbragaglia andF. Toschi, New J. Phys.
6
, 37 (2004).[15] In Ref. [6] the authors argued that Eqs. (1) and (2), with
−
1
< β <
0, could lead to
genuine multifractality
. Theirresults, obtained by a pseudospectral DNS with a ﬁniteviscosity, had a scaling range far smaller than can beachieved now. Our results for values of
β
in this rangewill be reported elsewhere.[16] This condition is opposite to the Courant–Friedrich–Lewytype condition, which is not needed with our algorithm[17] Deviations from this powerlaw r´egime occur at smallvalues of
r
because of the ultraviolet cutoﬀ Λ for thestochastic force (2).[18] Details on such questions can be found in theearlier version of the present paper, available at
http://xxx.lanl.gov/abs/nlin.CD/0406049v1
[19] To estimate shock locations we look at groups of four gridpoints where the discretized velocity gradient changes itssign twice (they correspond to a “zigzag” in the velocityproﬁle).[20] Similar plots from lowerresolution viscous spectral simulations were obtained in Ref. [5].