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Large-scale multifractality and lack of self-similar decay for Burgers and three-dimensional Navier–Stokes turbulence
Published online by Cambridge University Press: 07 August 2025
Abstract
We study decaying turbulence in the one-dimensional (1-D) Burgers equation (Burgulence) and 3-D Navier–Stokes (NS) turbulence. We first investigate the decay in time 
$t$ of the energy 
$E(t)$ in Burgulence, for a fractional Brownian initial potential, with Hurst exponent 
$H$, and demonstrate rigorously a self-similar time decay of 
$E(t)$, previously determined heuristically. This is a consequence of the non-trivial boundedness of the energy for any positive time. We define a spatially forgetful oblivious fractional Brownian motion (OFBM), with Hurst exponent 
$H$, and prove that Burgulence, with an OFBM as initial potential 
$\varphi _0(x)$, is not only intermittent, but it also displays a, hitherto unanticipated, large-scale bifractality or multifractality; the latter occurs if we combine OFBMs, with a distribution of 
$H\hbox{-}$values. This is the first rigorous proof of genuine multifractality for turbulence in a nonlinear hydrodynamical partial differential equation. We then present direct numerical simulations (DNSs) of freely decaying turbulence, capturing some aspects of this multifractality. For Burgulence, we investigate such decay for two cases: (a) 
$\varphi _0(x)$ a multifractal random walk that crosses over to a fractional Brownian motion beyond a cross-over scale 
$\mathcal{L}$, tuned to go from small- to large-scale multifractality; (b) initial energy spectra 
$E_0(k)$, with wavenumber 
$k$, having one or more power-law regions, which lead, respectively, to self-similar and non-self-similar energy decay. Our analogous DNSs of the 3-D NS equations also uncover self-similar and non-self-similar energy decay. Challenges confronting the detection of genuine large-scale multifractality, in numerical and experimental studies of NS and Magnetohydrodynamics turbulence, are highlighted.
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References
Alexakis, A. & Biferale, L. 2018 Cascades and transitions in turbulent flows. Phys. Rep. 767, 1–101.10.1016/j.physrep.2018.08.001CrossRefGoogle Scholar
Avellaneda, M. & Weinan, E. 1995 Statistical properties of shocks in Burgers turbulence. Commun. Math. Phys. 172 (1), 13–38.Google Scholar
Bacry, E., Delour, J. & Muzy, J.F. 2001
a Multifractal random walk. Phys. Rev. E
64 (2), 026103.10.1103/PhysRevE.64.026103CrossRefGoogle ScholarPubMed
Bacry, E., Delour, J. & Muzy, J.F. 2001
b Modelling financial time series using multifractal random walks. Physica A 299, 84–92.10.1016/S0378-4371(01)00284-9CrossRefGoogle Scholar
Barenblatt, G.I. 1996 Scaling, Self-Similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics. Cambridge University Press.CrossRefGoogle Scholar
Barenblatt, G.I. & Zel’dovich, Y.B. 1972 Self-similar solutions as intermediate asymptotics. Annu. Rev. Fluid Mech. 4 (1), 285–312.Google Scholar
Batchelor, G.K. & Townsend, A.A. 1947 Decay of vorticity in isotropic turbulence. Proc. R. Soc. Lond. A. Math. Phys. Sci. 190, 534–550.Google Scholar
Benzi, R., Biferale, L., Crisanti, A., Paladin, G., Vergassola, M. & Vulpiani, A. 1993 A random process for the construction of multiaffine fields. Physica D: Nonlinear Phenom. 65 (4), 352–358.Google Scholar
Biferale, L., Boffetta, G., Celani, A., Lanotte, A., Toschi, F. & Vergassola, M. 2003 The decay of homogeneous anisotropic turbulence. Phys. Fluids 15 (8), 2105–2112.10.1063/1.1582859CrossRefGoogle Scholar
Birkhoff, G. 1954 Fourier synthesis of homogeneous turbulence. Commun. Pure Appl. Maths 7 (1), 19–44.Google Scholar
Borue, V. & Orszag, S.A. 1995 Self-similar decay of three-dimensional homogeneous turbulence with hyperviscosity. Phys. Rev. E
51 (2), R856–R859.Google ScholarPubMed
Canuto, C., Hussaini, M.Y., Quarteroni, A. & Zang, T. 2007 Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer Science & Business Media.Google Scholar
Chaves, M., Eyink, G., Frisch, U. & Vergassola, M. 2001 Universal decay of scalar turbulence. Phys. Rev. Lett. 86 (11), 2305–2308.CrossRefGoogle ScholarPubMed
Cole, J.D. 1951 On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Maths 9 (3), 225–236.CrossRefGoogle Scholar
Comte-Bellot, G. & Corrsin, S. 1966 The use of a contraction to improve the isotropy of grid-generated turbulence. J. Fluid Mech. 25 (4), 657–682.CrossRefGoogle Scholar
Daley, D.J. & Vere-Jones, D. 2006 An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods. Springer Science & Business Media.Google Scholar
Davidson, P.A. 2023 The decay of energy and scalar variance in axisymmetric turbulence. Atmosphere-BASEL 14 (6), 1019.10.3390/atmos14061019CrossRefGoogle Scholar
Davidson, P.A., Okamoto, N. & Kaneda, Y. 2012 On freely decaying, anisotropic, axisymmetric Saffman turbulence. J. Fluid Mech. 706, 150–172.10.1017/jfm.2012.242CrossRefGoogle Scholar
De, S., Mitra, D. & Pandit, R. 2023 Dynamic multiscaling in stochastically forced Burgers turbulence. Sci. Rep. 13 (1), 7151.Google ScholarPubMed
De, S., Mitra, D. & Pandit, R. 2024 Uncovering the multifractality of Lagrangian pair dispersion in shock-dominated turbulence. Phys. Rev. Res. 6 (2), L022032.CrossRefGoogle Scholar
Dietrich, C.R. & Newsam, G.N. 1997 Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. Siam J. Sci. Comput. 18 (4), 1088–1107.10.1137/S1064827592240555CrossRefGoogle Scholar
Eyink, G.L. & Thomson, D.J. 2000 Free decay of turbulence and breakdown of self-similarity. Phys. Fluids 12 (3), 477–479.10.1063/1.870279CrossRefGoogle Scholar
Eyink, G.L. & Xin, J. 2000 Self-similar decay in the Kraichnan model of a passive scalar. J. Stat. Phys. 100 (3), 679–741.Google Scholar
Falkovich, G., Gawȩdzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73 (4), 913–975.CrossRefGoogle Scholar
Feller, W. 1991 An Introduction to Probability Theory and Its Applications, Volume 2, vol. 81. John Wiley & Sons.Google Scholar
Frisch, U. 1995 Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press.Google Scholar
Frisch, U. & Bec, J. 2002 Burgulence. In New Trends in Turbulence Turbulence: Nouveaux Aspects: 31 July–1 September 2000, pp. 341–383, Springer.Google Scholar
Frisch, U., Kurien, S., Pandit, R., Pauls, W., Ray, S.S., Wirth, A. & Zhu, J.-Z. 2008 Hyperviscosity, Galerkin truncation, and bottlenecks in turbulence. Phys. Rev. Lett. 101 (14), 144501.CrossRefGoogle ScholarPubMed
Frisch, U., Lesieur, M. & Schertzer, D. 1980 Comments on the quasi-normal Markovian approximation for fully-developed turbulence. J. Fluid Mech. 97 (1), 181–192.10.1017/S0022112080002492CrossRefGoogle Scholar
Frisch, U., Pouquet, A., Léorat, J. & Mazure, A. 1975 Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence. J. Fluid Mech. 68 (4), 769–778.CrossRefGoogle Scholar
Girsanov, I.V. 1960 On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theor. Probab. Applics. 5 (3), 285–301.10.1137/1105027CrossRefGoogle Scholar
Groeneboom, P. 1983 The concave majorant of Brownian motion. Ann. Probab. 11 (4), 1016–1027.Google Scholar
Gurbatov, S.N., Simdyankin, S.I., Aurell, E., Frisch, U. & Tóth, G. 1997 On the decay of Burgers turbulence. J. Fluid Mech. 344, 339–374.10.1017/S0022112097006241CrossRefGoogle Scholar
Hairer, E., Nørsett, S.P. & Wanner, G. 1993 Solving Ordinary Differential Equations I (2nd Revised. Ed.): Nonstiff Problems. Springer-Verlag.Google Scholar
Hopf, E. 1948 A mathematical example displaying features of turbulence. Commun. Pure Appl. Maths 1 (4), 303–322.CrossRefGoogle Scholar
Hopf, E. 1950 The partial differential equation
$u_{t} + u u_{x} = \mu _{x x}$
. Commun. Pure Appl. Maths 3 (3), 201–230.Google Scholar
$u_{t} + u u_{x} = \mu _{x x}$
. Commun. Pure Appl. Maths 3 (3), 201–230.Google ScholarIshida, T., Davidson, P.A. & Kaneda, Y. 2006 On the decay of isotropic turbulence. J. Fluid Mech. 564, 455–475.Google Scholar
Kalelkar, C. & Pandit, R. 2004 Decay of magnetohydrodynamic turbulence from power–law initial conditions. Phys. Rev. E
69 (4), 046304.10.1103/PhysRevE.69.046304CrossRefGoogle ScholarPubMed
Kida, S. 1979 Asymptotic properties of Burgers turbulence. J. Fluid Mech. 93 (2), 337–377.10.1017/S0022112079001932CrossRefGoogle Scholar
Kida, S. & Goto, S. 1997 A Lagrangian direct-interaction approximation for homogeneous isotropic turbulence. J. Fluid Mech. 345, 307–345.Google Scholar
Kolmogorov, A.N. 1941 On degeneration of isotropic turbulence in an incompressible viscous liquid. Dokl. Akad. Nauk SSSR 31, 538–540.Google Scholar
Kraichnan, R.H. 1968 Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11 (5), 945–953.10.1063/1.1692063CrossRefGoogle Scholar
Krogstad, P.-Å. & Davidson, P.A. 2011 Freely decaying, homogeneous turbulence generated by multi-scale grids. J. Fluid Mech. 680, 417–434.Google Scholar
Krstulovic, G. & Nazarenko, S. 2024 Initial evolution of three-dimensional turbulence en route to the Kolmogorov state: emergence and transformations of coherent structures, self-similarity, and instabilities. Phys. Rev. Fluids 9 (3), 034610.10.1103/PhysRevFluids.9.034610CrossRefGoogle Scholar
Lévy, P. 1953 Random Functions: General Theory with Special Reference to Laplacian Random Functions. vol. 1. University of California Press.Google Scholar
Loitsiansky, L.G. 1939 Some basic laws for isotropic turbulent flow. C.A.H.I. (Moscow), Rept. 440 (1939); also N.A.C.A. Tech. Memo. (1945) 1079.Google Scholar
Mandelbrot, B.B., Van Ness, J.W. 1968 Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (4), 422–437.10.1137/1010093CrossRefGoogle Scholar
Marinoni, A. & Marani, P.C. 2004 Il codice Atlantico di Leonardo da Vinci, indice per materie e alfabetico. Giunti, Milan.Google Scholar
Meldi, M. & Sagaut, P. 2012 On non-self-similar regimes in homogeneous isotropic turbulence decay. J. Fluid Mech. 711, 364–393.10.1017/jfm.2012.396CrossRefGoogle Scholar
Molchan, G. 2017 The inviscid Burgers equation with fractional Brownian initial data: the dimension of regular Lagrangian points. J. Stat. Phys. 167 (6), 1546–1554.CrossRefGoogle Scholar
Molchan, G.M. 1997 Burgers equation with self-similar Gaussian initial data: tail probabilities. J. Stat. Phys. 88 (5-6), 1139–1150.CrossRefGoogle Scholar
Molchan, G.M. 2000 On the maximum of a fractional Brownian motion. Theor. Probab. Applics 44 (1), 97–102.10.1137/S0040585X97977379CrossRefGoogle Scholar
Noullez, A., Gurbatov, S.N., Aurell, E. & Simdyankin, S.I. 2005 Global picture of self-similar and non-self-similar decay in Burgers turbulence. Phys. Rev. E
71 (5), 056305.10.1103/PhysRevE.71.056305CrossRefGoogle ScholarPubMed
Noullez, A. & Vergassola, M. 1994 A fast Legendre transform algorithm and applications to the adhesion model. J. Sci. Comput. 9 (3), 259–281.10.1007/BF01575032CrossRefGoogle Scholar
Panickacheril John, J., Donzis, D.A. & Sreenivasan, K.R. 2022 Laws of turbulence decay from direct numerical simulations. Phil. Trans. R. Soc. A
380 (2218), 20210089.CrossRefGoogle ScholarPubMed
Parisi, G. & Frisch, U. 1985 On the singularity structure of fully developed turbulence. In Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics in Proceedings of the International School of Physics ‘E. Fermi’ 1983, Varenna, Italy, pp. 84–88.Google Scholar
Piterbarg, V. & Prisyazhnyuk, V. 1978 Asymptotics of the probability of large deviation for a nonstationary Gaussian process. Prob. Theor. Math. Stat. 18, 121–134.Google Scholar
Pitman, J.W. 1983 Remarks on the Convex Minorant of Brownian Motion. In Seminar on Stochastic Processes, 1982, (ed. E. Cinlar, K.L. Chung & R.K. Getoor), vol. 5, pp. 219–227.Google Scholar
Proudman, I. & Reid, W.H. 1954 On the decay of a normally distributed and homogeneous turbulent velocity field. Phil. Trans. R. Soc. Lond. A
247, 163–189.Google Scholar
Rowan, K. 2024 On anomalous diffusion in the Kraichnan model and correlated-in-time variants. Arch. Ration. Mech. Anal. 248 (5), 93.10.1007/s00205-024-02045-0CrossRefGoogle Scholar
Roy, D. 2021 Steady state properties of discrete and continuous models of nonequilibrium phenomena. PhD thesis, Institute of Science.Google Scholar
Saffman, P.G. 1967 The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27 (3), 581–593.Google Scholar
Sankar Ray, S., Mitra, D. & Pandit, R. 2008 The universality of dynamic multiscaling in homogeneous, isotropic Navier–Stokes and passive-scalar turbulence. New J. Phys. 10 (3), 033003.10.1088/1367-2630/10/3/033003CrossRefGoogle Scholar
She, Z.-S., Aurell, E. & Frisch, U. 1992 The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148 (3), 623–641.Google Scholar
Son, D.T. 1999 Turbulent decay of a passive scalar in the Batchelor limit: exact results from a quantum-mechanical approach. Phys. Rev. E
59 (4), R3811–R3814.Google Scholar
Stalp, S.R., Skrbek, L. & Donnelly, R.J. 1999 Decay of grid turbulence in a finite channel. Phys. Rev. Lett. 82 (24), 4831–4834.Google Scholar
Tatsumi, T., Kida, S. & Mizushima, J. 1978 The multiple-scale cumulant expansion for isotropic turbulence. J. Fluid Mech. 85 (1), 97–142.10.1017/S0022112078000555CrossRefGoogle Scholar
Valente, P.C. & Vassilicos, J.C. 2011 The decay of turbulence generated by a class of multiscale grids. J. Fluid Mech. 687, 300–340.Google Scholar
Vergassola, M., Dubrulle, B., Frisch, U. & Noullez, A. 1994 Burgers’ equation, devil’s staircases and the mass distribution for large-scale structures. Astron. Astrophys. 289 (2), 325–356.Google Scholar
von Kármán, T. & Howarth, L. 1938 On the statistical theory of isotropic turbulence. Proc. R. Soc. Lond. A, Math. Phys. Sci. 164 (917), 192–215.Google Scholar
Wilczek, M. 2016 Non-gaussianity and intermittency in an ensemble of Gaussian fields. New J. Phys. 18 (12), 125009.10.1088/1367-2630/18/12/125009CrossRefGoogle Scholar