Abstract
Extreme atmospheric events are intimately related to the statistics of atmospheric turbulent velocities. These, in turn, exhibit multifractal scaling, which is determining the nature of the asymptotic behavior of velocities, and whose parameter evaluation is therefore of great interest currently. We combine singular value decomposition techniques and wavelet transform analysis to generalize the multifractal formalism to vector-valued random fields. The so-called Tensorial Wavelet Transform Modulus Maxima (TWTMM) method is calibrated on synthetic self-similar 2D vector-valued multifractal measures and monofractal 3D vector-valued fractional Brownian fields. We report the results of some application of the TWTMM method to turbulent velocity and vorticity fields generated by direct numerical simulations of the incompressible Navier–Stokes equations. This study reveals the existence of an intimate relationship \({D_{{\bf v}}(h+1)=D_{\varvec{\omega}}(h)},\) between the singularity spectra of these two vector fields which are found significantly more intermittent than previously estimated from longitudinal and transverse velocity increment statistics.
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References
Abry P, Fauve S, Flandrin P, Laroche C (1994) J Phys II Fr 4:725
Antonia RA, Pearson BR (1997) Europhys Lett 40:123
Arad I, Dhruva B, Kurien S, L’vov VS, Procaccia I, Sreenivasan KR (1998) Phys Rev Lett 81:5330
Arad I, L’vov VS, Procaccia I (1999) Phys Rev E 59:6753
Argoul F, Arneodo A, Elezgaray J, Grasseau G, Murenzi R (1990) Phys Rev A 41:5537
Arneodo A, Argoul F, Bacry E, Muzy JF, Tabard M (1992a) Phys Rev Lett 68:3456
Arneodo A, Argoul F, Muzy JF, Tabard M (1992b) Phys Lett A 171:31; Phys A 188:217
Arneodo A, Argoul F, Muzy JF, Tabard M, Bacry E (1993) Fractals 1:629
Arneodo A, Bacry E, Muzy JF (1995a) Phys A 213:232
Arneodo A, Bacry E, Muzy JF (1995b) Phys Rev Lett 74:4823
Arneodo A, Argoul F, Bacry E, Elezgaray J, Muzy JF (1995c) Ondelettes, Multifractales et Turbulences: de l’ADN aux croissances cristallines. Diderot Editeur, Art et Sciences, Paris
Arneodo A, Bacry E, Graves PV, Muzy JF (1995d) Phys Rev Lett 74:3293
Arneodo A, d’Aubenton-Carafa Y, Bacry E, Graves PV, Muzy JF, Thermes C (1996) Phys D 96:291
Arneodo A, Bacry E, Jaffard S, Muzy JF (1997a) J Stat Phys 87:179; J Fourier Anal Appl 4:159 (1998); In: CRM proceedings and lecture notes 18:315 (1999)
Arneodo A, Muzy JF, Roux SG (1997b) J Phys II Fr 7:363
Arneodo A, Manneville S, Muzy JF (1998a) Eur Phys J B 1:129
Arneodo A, Muzy JF, Sornette D (1998b) Eur Phys J B 2:277
Arneodo A, Manneville S, Muzy JF, Roux SG (1999a) Phil Trans R Soc Lond A 357:2415
Arneodo A, Decoster N, Roux SG (1999b) Phys Rev Lett 83:1255; Eur Phys J B 15:567 (2000)
Arneodo A, Audit B, Decoster N, Muzy JF, Vaillant C (2002) In: Bunde A, Kropp J, Schellnhuber HJ (eds) The science of disasters: climate disruptions, heart attacks and market crashes. Springer, Berlin, p 27
Arneodo A, Decoster N, Kestener P, Roux SG (2003) Adv Imaging Electr Phys 126:1
Arrault J, Arneodo A, Davis A, Marshak A (1997) Phys Rev Lett 79:75
Audit B, Thermes C, Vaillant C, d’Aubenton-Carafa Y, Muzy JF, Arneodo A (2001) Phys Rev Lett 86:2471
Audit B, Vaillant C, Arneodo A, d’Aubenton-Carafa Y, Thermes C (2002a) J Mol Biol 316:903
Audit B, Bacry E, Muzy JF, Arneodo A (2002b) IEEE Trans Info Theory 48:2938
Bacry E, Muzy JF, Arneodo A (1993) J Stat Phys 70:635
Badii R, Politi A (1984) Phys Rev Lett 52:1661; J Stat Phys 40:725 (1985)
Barabási AL, Vicsek T (1991) Phys Rev A 44:2730
Barabási AL, Széfalusy P, Vicsek T (1991) Phys A 178:17
Baudet C, Michel O, Williams WJ (1999) Phys D 128:1
Belin F, Maurer J, Tabeling P, Willaime H (1996) J Phys II Fr 6:1
Bendjoudi H, Hubert P, Schertzer D, Lovejoy S (1997) Geosci Surf Hydrol 325:323
Benzi R, Ciliberto S, Baudet C, Chavarria GR (1995) Phys D 80:385
Benzi R, Struglia R, Tripiccione R (1996) Phys Rev E 53:R5565
Bergé P, Pomeau Y, Vidal C (1986) Order within Chaos. Wiley, New York
Bershadskii A, Nakano T, Fukayama D, Gotoh T (2000) Eur Phys J B 18:95
Boratov ON, Pelz RB (1997) Phys Fluids 9:1400
Borue V, Orszag SA (1996) Phys Rev E 53:R21
Brachet ME (1991) Fluid Dyn Res 8:1
Burlando P, Rosso R (1996) J Hydrol 187:45
Cadot O, Douady S, Couder Y (1995) Phys Fluids 7:630
Camussi R, Benzi R (1997) Phys Fluids 9:257
Camussi R, Guj G (1997) J Fluid Mech 348:177
Castaing B, Gagne Y, Marchand M (1993) Phys D 68:387
Castro JJ, Carsteanu AA, Flores CG (2004) Phys A 33:206
Chanal O, Chabaud B, Castaing B, Hebral B (2000) Eur Phys J B 17:309
Chen S, Doolen GD, Kraichnan RH, She ZS (1992) Phys Fluids A 5:458
Chen S, Doolen GD, Kraichnan RH, Wang LP (1995) Phys Rev Lett 74:1755
Chen S, Sreenivasan KR, Nelkin M (1997a) Phys Rev Lett 79:1253
Chen S, Sreenivasan KR, Nelkin M, Cao N (1997b) Phys Rev Lett 79:2253
Chevillard L (2004) Ph.D. thesis, University of Bordeaux I
Chevillard L, Castaing B, Lévêque E (2005) Eur Phys J B 45:561
Chevillard L, Roux SG, Lévêque E, Mordant N, Pinton JF, Arneodo A (2003) Phys Rev Lett 91:214502
Collet P, Lebowitz J, Porzio A (1987) J Stat Phys 47:609
Cuypers Y, Maurel A, Petitjeans P (2003) Phys Rev Lett 91:194502
Cvitanovic P (ed) (1984) Universality in Chaos, Hilger, Bristol
Daubechies I (1992) Ten lectures on wavelets. SIAM, Philadelphia
De Michele C, Kottegoda NT, Rosso R (2002) Water Sci Technol 45(2):83
Decoster N, Roux SG, Arneodo A (2000) Eur Phys J B 15:739
Delour J, Muzy JF, Arneodo A (2001) Eur Phys J B 23:243
Dernoncourt B, Pinton JF, Fauve S (1998) Phys D 117:181
Dhruva B, Tsuji Y, Sreenivasan KR (1997) Phys Rev E 56:R4928
Douady A, Couder Y, Brachet ME (1991) Phys Rev Lett 67:983
Falconer KJ, O’Neil TC (1996) Proc R Soc Lond A 452:1433
Farge M, Pellegrino G, Schneider K (2001) Phys Rev Lett 87:054501
Fauve S, Laroche C, Castaing B (1993) J Phys II Fr 3:271
Frisch U (1995) Turbulence. Cambridge University Press, Cambridge
Gagne Y, Marchand M, Castaing B (1994) J Phys II Fr 4:1
Golub GH, Loan CV (1989) Matrix computations, 2nd edn. John Hopkins University Press, Baltimore
Goupillaud P, Grossmann A, Morlet J (1984) Geoexploration 23:85
Grassberger P, Procaccia I (1983) Phys Rev Lett 50:346; Phys D 9:189
Grassberger P, Badii R, Politi A (1988) J Stat Phys 51:135
Grossmann A, Morlet J (1984) SIAM J Math Anal 15:723
Grossmann S, Lohse D, Reeh A (1997) Phys Fluids 9:3817
Halsey TC, Jensen MH, Kadanoff LP, Procaccia I, Schraiman BI (1986) Phys Rev A 33:1141
Hao BL (ed.) (1984) Chaos, World Scientific, Singapore
He G, Chen S, Kraichnan RH, Zhang R, Zhou Y (1998) Phys Rev Lett 81:4636
Hentschel HGE (1994) Phys Rev E 50:243
Hosokawa I, Yamamoto K (1992) Phys Fluids A 4:457
Hosokawa I, Oide S, Yamamoto K (1996) Phys Rev Lett 77:4548
Hubert P, Tessier Y, Lovejoy S, Schertzer D, Schmitt F, Ladoy P, Carbonnel JP, Violette S, Desurosne I (1993) Geophys Res Lett 20(10):931
Ivanov PC, Rosenblum MG, Peng CK, Mietus J, Havlin S, Stanley HE, Goldberger AL (1996) Nature 383:323
Ivanov PC, Nunes Amaral LA, Goldberger AL, Havlin S, Rosenblum MG, Struzik ZR, Stanley HE (1999) Nature 399:461
Jaffard S (1997) SIAM J Math Anal 28:944
Jimenez J, Wray AA, Saffman PG, Rogallo RS (1993) J Fluid Mech 255:65
Kerr R (1985) J Fluid Mech 153:31
Kestener P (2003) Ph.D. thesis, University of Bordeaux I
Kestener P, Arneodo A (2003) Phys Rev Lett 91:194501; In: Prenel J, Bailly Y (eds) Proceedings of the 4th pacific symposium on flow visualization and image processing (to appear)
Kestener P, Arneodo A (2004) Phys Rev Lett 93:044 501
Kestener P, Lina JM, Saint-Jean P, Arneodo A (2001) Image Anal Stereol 20:169
Kevlahan NK-R, Vassilicos JC (1994) Proc R Soc Lond A 447:341
Khalil A, Joncas G, Nekka F, Kestener P, Arneodo A (2004) Astrophysical Journal Supplement Series 165:512
Khun A, Argoul F, Muzy JF, Arneodo A (1994) Phys Rev Lett 73:2998
Kolmogorov AN (1941) C R Acad Sci USSR 30:301
Kolmogorov AN (1962) J Fluid Mech 13:82
L’vov VS, Procaccia I (1996) Phys Fluids 8:2565
Malécot Y, Auriault C, Kahalerras H, Gagne Y, Chanal O, Chabaud B, Castaing B (2000) Eur Phys J B 16:549
Mallat S (1998) A wavelet tour in signal processing. Academic, New York
Mallat S, Hwang WL (1992) IEEE Trans Inform Theory 38:617
Mallat S, Zhong S (1992) IEEE Trans Pattern Anal Mach Intell 14:710
Mandelbrot BB (1974) J Fluid Mech 62:331
Mandelbrot BB (1977) Fractals: form, chance and dimensions (Freeman, San Francisco); The fractal geometry of nature (Freeman, San Francisco, 1982)
Mandelbrot BB (1989) Fractals and multifractals: noise, turbulence and galaxies, vol 1, Selecta. Springer, Berlin
Meisel LV, Jonhson M, Cote PJ (1992) Phys Rev A 45:6989
Meneveau C, Sreenivasan KR (1991) J Fluid Mech 224:429
Metais O, Lesieur M (1992) J Fluid Mech 239:157
Meyer Y (1990) Ondelettes. Herman, Paris
Molenaar J, Herweijer J, Van de Water W (1995) Phys Rev E 52:496
Monin AS, Yaglom AM (1975) Statistical fluid mechanics, vol 2. MIT Press, Cambridge
Muzy JF, Bacry E, Arneodo A (1991) Phys Rev Lett 67:3515; Int J Bifurc Chaos 4:245 (1994)
Muzy JF, Bacry E, Arneodo A (1993) Phys Rev E 47:875
Muzy JF, Sornette D, Delour J, Arneodo A (2000) Quant Finance 1:131
Nelkin M (1999) Phys Fluids 11:2202
Nicolleau F, Vassilicos JC (1999) Phil Trans R Soc Lond A 357:2439
Obukhov AM (1962) J Fluid Mech 13:77
O’Neil J, Meneveau C (1993) Phys Fluids A 5:158
Paladin G, Vulpiani A (1987) Phys Rep 156:148
Parisi G, Frisch U (1985) In: Turbulence and predictability in geophysical fluid dynamics and climate dynamics; In: Ghil M, Benzi R, Parisi G (eds) Proceedings of International School. North-Holland, Amsterdam, p 84
Pedrizzetti G, Novikov EA, Praskovsky AA (1996) Phys Rev E 53:475
Peitgen HO, Saupe D, Eds (1987) The science of fractal images. Springer, New York
Praskovsky AA (1992) Phys Fluids A 4:2589
Praskovsky AA, Oncley S (1994) Europhys Lett 28:635
Pumir A (1993) Phys Fluids 6:2071
Rand D (1989) Ergod Th Dyn Sys 9:527
Roux SG (1996) Ph.D. thesis, University of Aix-Marseille II
Roux SG, Muzy JF, Arneodo A (1999) Eur Phys J B 8:301
Roux SG, Arneodo A, Decoster N (2000) Eur Phys J B 15:765
Schuster HG (1984) Deterministic chaos. Physik Verlag, Weimheim
She ZS, Jackson E, Orszag SA (1990) Nature 344:226; Proc R Soc Lond 434:101 (1991)
Siggia ED (1981) J Fluid Mech 107:375
Stolovitzky G, Sreenivasan KR (1994) Rev Mod Phys 66:229
Stolovitzky G, Kailasnath P, Sreenivasan KR (1992) Phys Rev Lett 69:1178
Tanaka M, Kida S (1993) Phys Fluids A 5:2079
Thoroddsen ST (1995) Phys Fluids 7:691
Thoroddsen ST, Van Atta CW (1992) Phys Fluids A 4:2592
Van de Water W, Herweijer J (1996) Bull Am Phys Soc 41:1782
Vassilicos JC, Hunt JC (1991) Proc R Soc Lond 435:505
Veneziano D, Furcolo P (2002) Water Resour Res 38(12):1306
Villermaux E, Sixou B, Gagne Y (1995) Phys Fluids 7:2008
Vincent A, Meneguzzi M (1991) J Fluid Mech 225:1, 258:245 (1994)
Wang LP, Chen S, Brasseur JG, Wyngaard JC (1996) J Fluid Mech 309:113
Acknowledgments
We are very endebted to E. Lévêque for allowing us to have access to his DNS numerical data and to the CNRS under GDR turbulence.
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Kestener, P., Arneodo, A. A multifractal formalism for vector-valued random fields based on wavelet analysis: application to turbulent velocity and vorticity 3D numerical data. Stoch Environ Res Risk Assess 22, 421–435 (2008). https://doi.org/10.1007/s00477-007-0121-6
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DOI: https://doi.org/10.1007/s00477-007-0121-6