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Is Multiscaling an Artifact in the Stochastically Forced Burgers Equation?

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Is Multiscaling an Artifact in the Stochastically Forced Burgers Equation?
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    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 one-dimensional Burgers equation with a white-in-time, Gaussianrandom force that has a Fourier-space spectrum ∼ 1 /k , where  k  is the wave number. From very-high-resolution 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, 05.45-a, 05.40-a Homogeneous, isotropic fluid turbulence is often char-acterized 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 de-note 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, exper-iments and direct numerical simulations (DNS) suggest multiscaling   with  ζ   p  a nonlinear, monotonically increas-ing, 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 well-known difficulties [3]. It isimportant therefore to establish, or rule out, multiscal-ing of structure functions in simpler forms of turbulence,such as passive-scalar, passive-vector or Burgers turbu-lence. Significant progress, both analytical and numeri-cal, has been made in confirming multiscaling in passive-scalar and passive-vector problems (see, e.g., Ref. [4] fora review). The linearity of the passive-scalar and passive-vector 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 one-dimensional, Burgers equa-tion with stochastic self-similar forcing, studied earlierin Refs. [5, 6]. It is by far the simplest  nonlinear   partialdifferential equation (PDE) that has the potential to dis-play 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 zero-mean, space-periodic Gaus-sian 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 re-strict ourselves to the case  β   =  − 1 and assume spatial pe-riodicity 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 exhibitself-similar scaling with exponents  p/ 3 and implies a K41-type  − 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 simi-lar to that observed when the Burgers equation is forcedonly at large spatial scales [7, 8]. We overcome the limitations of these earlier stud-ies [5, 6] by adapting the algorithm of Refs. [9, 10] to develop a state-of-the-art technique for the numerical so-lution 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 power-law 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 small-scale fluctuations much stronger than those present in the force. Log-log 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 least-squares 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 white-in-time force is approached by shotnoise. Between successive kicks we evolve the velocityby using the following well-known result on the solutionsto the unforced Burgers equation in the limit of vanish-ing 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 dy-namics, colliding Lagrangian particles form shocks anddo not cross each other. At small scales we want to haveunforced Burgers dynamics with well-identifiable shocks.At least four mesh points are needed for unambiguousidentification of a shock; since the maximum wavenum-ber is  N/ 2, we set ˆ f  ( k,t ) = 0 beyond an ultra-violetcutoff 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 in-dependent Gaussian random functions with zero meanand a Fourier-space 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 char-acteristic velocity difference at large length scales  O ( L ).The first ensures that a typical Lagrangianparticle movesat least half the mesh-size in time  δt  (otherwise, it wouldstay put)[16]; the second, which expresses that  δt  is com-parable to the turnover times at the scale  L/ Λ, guaran-tees that, at scales larger than  L/ Λ the time-step,  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 good-quality 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 large-eddy-turnovertime. Data from  T  tr  time steps are discarded so that tran-sients 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 per-formed 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 log-log plots of Figs. (1 b) and (1 c) of   S  abs  p  ( r ) for  p  = 3 and 4 showpower-law 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 least-squaresfits to all triplets of consecutive points deep inside thepower-law 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 thebifractal-scaling prediction (full lines). As we shall seethis deviation need not necessarily imply “multiscaling”for the structure functions. We have also considered thepossibility of already well-understood artifacts, such asthe role of temporal transients [12] and finite-size effectswhich can round sharp bifractal transitions [13], and de-cided that they do not play any major role in the presentproblem [18].Consider  P  c ( s ), the cumulative probability distribu-tion function of shock strengths  s . Simple scaling argu-ments 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 argu-ment favors  γ   =  − 3 : the total input energy,  E  in  =    Λ k 0 D ( k ) dk  ∼  lnΛ, where Λ is the ultra-violet cutoff.In the limit of vanishing viscosity, the energy dissipa-tion 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 dis-tribution 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 least-squares 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 “anoma-lous” exponents in Fig. (2 a) might well be suspect.To explore this further, consider the third-order struc-ture 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 rela-tion in fluid turbulence by a simple generalization of theproof given in Ref. [9] for the Burgers equation forced de-terministically 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 log-log 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 prob-ably 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 log-log 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 ex-ponent 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 suit-ably chosen factor  λ . In a log-log 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 posi-tive 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 ) ear-lier. Indeed the negative increments are dominated bythe contribution from shocks. Just as  P  c ( s ) has cut-offs  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 sub-leading term ( r 4 / 3 ) is crucial. It im-plies that, for any finite  r , a naive power-law fit to  S  4  canyield a scaling exponent less than unity. The presence of sub-leading, power-law terms with opposite signs also ex-plains the small apparent “anomalous” scaling behaviorobserved for other values of   p  in our simulations. A simi-lar artifact involving two competing power-laws has beendescribed in Ref. [14].In conclusion, we have performed very-high-resolutionnumerical simulations of the stochastically forced Burg-ers equation with a 1 /k  forcing spectrum. A naive inter-pretation of our data shows apparent multiscaling phe-nomenon. But our detailed analysis has identified ahitherto-unknown 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) Log-log plots of the cumulative probability distribution function  P  c ( s ) versusshock strengths  s  obtained from an average over 1000 snapshots. A least-squares 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 simple-scaling prediction  γ   =  − 3 is indicated by the straightline. The inset shows log-log 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) Log-log plots of   S  3 ( r ) (crosses), S  abs3  ( r ) (dashed line) and   ( δ  + u ) 3   (squares) versus  r . The continuous line is a least-square 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 log-log scale. Thedashed line is the right-hand side of Eq. (6); the left-hand side of this equation has been obtained for  N   = 2 20 ( ◦ ) (run B1). caling can masquerade as multiscaling. Our work illus-trates that the elucidation of multiscaling in spatiallyextended nonlinear systems, including the Navier–Stokesequation, requires considerable theoretical insight thatmust supplement state-of-the-art numerical simulationsand experiments.We thank E. Aurell, L. Biferale, A. Lanotte, and V.Yakhot for discussions. This research was supported bythe Indo-French Centre for the Promotion of AdvancedResearch (Project 2404-2), by CSIR (India), and by theEuropean Union under contracts HPRN-CT-2000-00162and HPRN-CT-2002-00300. 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. Ya-glom, F. David, and M. Lesieur eds., Les Houches SessionLXXIV (Springer EDP-Sciences, 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. 184-187. 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–Lewy-type condition, which is not needed with our al-gorithm[17] Deviations from this power-law 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 “zig-zag” in the velocityprofile).[20] Similar plots from lower-resolution viscous spectral sim-ulations were obtained in Ref. [5].
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