Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-16T21:44:40.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Physical mechanism of the inverse energy cascade of two-dimensional turbulence: a numerical investigation

Published online by Cambridge University Press:  25 January 2009

Z. XIAO ,
M. WAN ,
S. CHEN  and
G. L. EYINK
Z. XIAO
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, Peking University, Beijing 100871, P. R. China Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
M. WAN
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
S. CHEN
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, Peking University, Beijing 100871, P. R. China Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA Department of Applied Mathematics & Statistics, The Johns Hopkins University, Baltimore, MD 21218, USA Centre for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
G. L. EYINK
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA Department of Applied Mathematics & Statistics, The Johns Hopkins University, Baltimore, MD 21218, USA Centre for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We report an investigation of inverse energy cascade in steady-state two-dimensional turbulence by direct numerical simulation (DNS) of the two-dimensional Navier–Stokes equation, with small-scale forcing and large-scale damping. We employed several types of damping and dissipation mechanisms in simulations up to 20482 resolution. For all these simulations we obtained a wavenumber range for which the mean spectral energy flux is a negative constant and the energy spectrum scales as k−5/3, consistent with the predictions of Kraichnan (Phys. Fluids, vol. 439, 1967, p. 1417). To gain further insight, we investigated the energy cascade in physical space, employing a local energy flux defined by smooth filtering. We found that the inverse energy cascade is scale local, but that the strongly local contribution vanishes identically, as argued by Kraichnan (J. Fluid Mech., vol. 47, 1971, p. 525). The mean flux across a length scale ℓ was shown to be due mainly to interactions with modes two to eight times smaller. A major part of our investigation was devoted to identifying the physical mechanism of the two-dimensional inverse energy cascade. One popular idea is that inverse energy cascade proceeds via merger of like-sign vortices. We made a quantitative study employing a precise topological criterion of merger events. Our statistical analysis showed that vortex mergers play a negligible direct role in producing mean inverse energy flux in our simulations. Instead, we obtained with the help of other works considerable evidence in favour of a ‘vortex thinning’ mechanism, according to which the large-scale strains do negative work against turbulent stress as they stretch out the isolines of small-scale vorticity. In particular, we studied a multi-scale gradient (MSG) expansion developed by Eyink (J. Fluid Mech., vol. 549, 2006a, p. 159) for the turbulent stress, whose contributions to inverse cascade can all be explained by ‘thinning’. The MSG expansion up to second order in space gradients was found to predict well the magnitude, spatial structure and scale distribution of the local energy flux. The majority of mean flux was found to be due to the relative rotation of strain matrices at different length scales, a first-order effect of ‘thinning’. The remainder arose from two second-order effects, differential strain rotation and vorticity gradient stretching. Our findings give strong support to vortex thinning as the fundamental mechanism of two-dimensional inverse energy cascade.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 619 , 25 January 2009 , pp. 1 - 44
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Babiano, A. & Dubos, T. 2005 On the contribution of coherent vortices to the two-dimensional inverse energy cascade. J. Fluid Mech. 529, 97–116.Google Scholar
Basdevant, C. & Philipovitch, T. 1994 On the validity of the weiss criterion in two-dimensional turbulence. Phys. D 73, 17–30.Google Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12, 233–239.Google Scholar
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1 (5), 497–504.Google Scholar
Boffetta, G. 2007 Energy and enstrophy fluxes in the double cascade of 2d turbulence. J. Fluid Mech. 589, 253–260.CrossRefGoogle Scholar
Boffetta, G., Celani, A. & Vergassola, M. 2000 Inverse energy cascade in two-dimensional turbulence: deviations from gaussian behavior. Phys. Rev. E 61 (1), R29–R32.Google Scholar
Borue, V. 1994 Inverse energy cascade in stationary two-dimensional homogeneous turbulence. Phys. Rev. Lett. 72, 1475–1478.Google Scholar
Bouchet, F. 2003 Parameterization of two-dimensional turbulence using an anisotropic maximum entropy production principle. ArXiv:cond-mat/0305205.Google Scholar
Brasseur, J. G., Gong, H. & Chen, S. 1996 Subgrid-resolved scale dynamics in isotropic turbulence. In Advances in Turbulence VI (ed. Gavrilakis, S., Machiels, L. & Monkewitz, P. A.), pp. 201–204. Kluwer Academic Publishers.Google Scholar
Bruneau, C.-H., Fischer, P. & Kellay, H. 2007 The structures responsible for the inverse energy and the forward enstrophy cascades in two-dimensional turbulence. EPL 78 (3), 34002.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1987 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Carton, X., Maze, G. & Legras, B. 2002 A two-dimensional vortex merger in an external strain field. J. Turbul. 3, 045.CrossRefGoogle Scholar
Cerutti, S. & Meneveau, C. 1998 Intermittency and relative scaling of subgrid-scale energy dissipation in isotropic turbulence. Phys. Fluids A 10 (4), 928–937.Google Scholar
Charney, J. 1971 Geostrophic turbulence. J. Atmos. Sci. 28 (6), 1087–1095.Google Scholar
Chen, Q., Chen, S., Eyink, G. L. & Holm, D. D. 2003 a Intermittency in the joint cascade of energy and helicity. Phys. Rev. Lett. 90 (21), 214503.CrossRefGoogle ScholarPubMed
Chen, S., Ecke, R. E., Eyink, G. L., Rivera, M., Wan, M. & Xiao, Z. 2006 Physical mechanism of the two-dimensional inverse energy cascade. Phys. Rev. Lett. 96, 084502.CrossRefGoogle ScholarPubMed
Chen, S., Ecke, R. E., Eyink, G. L., Wang, X. & Xiao, Z. 2003 b Physical mechanism of the two-dimensional enstrophy cascade. Phys. Rev. Lett. 91, 214501.Google Scholar
Cho, J. & Lindborg, E. 2001 Horizontal velocity structure functions in the upper troposphere and lower stratosphere. 1. Observations. J. Geophys. Res. 106 (D10), 10223–10232.Google Scholar
Danilov, S. & Gurarie, D. 2001 Forced two-dimensional turbulence in spectral and physical space. Phys. Rev. E 63 (6), 061208.Google Scholar
Domaradzki, J. A. & Rogallo, R. S. 1990 Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence. Phys. Fluids A 2, 413–426.Google Scholar
Dritschel, D. G. 1993 Vortex properties of two-dimensional turbulence. Phys. Fluids A 5 (4), 984–997.Google Scholar
Dubos, T. & Babiano, A. 2002 Two-dimensional cascades and mixing: a physical space approach. J. Fluid Mech. 467, 81–100.Google Scholar
Eyink, G. L. 1994 Dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Phys. D 78, 222–240.Google Scholar
Eyink, G. L. 1996 Exact results on stationary turbulence in 2d: consequences of vorticity conservation. Phys. D 91, 97–142.Google Scholar
Eyink, G. L. 2005 Locality of turbulent cascades. Phys. D 207, 91–116.Google Scholar
Eyink, G. L. 2006 a Multiscale gradient expansion of the turbulent stress tensor. J. Fluid Mech. 549, 159–190.Google Scholar
Eyink, G. L. 2006 b A turbulent constitutive law for the two-dimensional inverse energy cascade. J. Fluid Mech. 549, 191–214.CrossRefGoogle Scholar
Falkovich, G. 1994 Bottleneck phenomenon in developed turbulence. Phys. Fluids 6, 1411–1414.Google Scholar
Frisch, U. & Sulem, P. L. 1984 Numerical simulation of the inverse cascade in two-dimensional turbulence. Phys. Fluids 27, 1921–1923.Google Scholar
Gage, K. S. 1979 Evidence for a k −5/3 law inertial range in mesoscale two-dimensional turbulence. J. Atmos. Sci. 36 (10), 1950–1957.2.0.CO;2>CrossRefGoogle Scholar
Gage, K. S. & Nastrom, G. D. 1986 Theoretical interpretation of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft during gasp. J. Atmos. Sci. 43 (7), 729–740.Google Scholar
Galperin, B., Nakano, H., Huang, H.-P. & Sukoriansky, S. 2004 The ubiquitous zonal jets in the atmospheres of giant planets and earth's oceans. Geo. Res. Lett. 31 (13), L13303.Google Scholar
Germano, M. 1992 Turbulence: the filtering approach. J. Fluid Mech. 238, 325–336.Google Scholar
Helman, J. L. & Hesselink, L. 1991 Visualizing vector field topology in fluid flows. IEEE Comp. Grap. Appl. 11 (3), 36–46.Google Scholar
Holloway, G. 1989 Subgridscale representation. In Oceanic Circulation Models: Combining Data and Dynamics (ed. Anderson, D. L. T. & Willebrand, J.), NATO. ASI Series, Series C, vol. 284, pp. 513–593. Kluwer Academic Publishers.CrossRefGoogle Scholar
Hua, B. L. & Klein, P. 1998 An exact criterion for the stirring properties of nearly two-dimensional turbulence. Phys. D 113, 98–110.Google Scholar
Irion, R. 1999 Fluid dynamics – soap films reveal whirling worlds of turbulence. Science 284 (5420), 1609–1610.Google Scholar
Kida, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Jpn. 50 (10), 3517–3520.CrossRefGoogle Scholar
Kimura, Y. & Herring, J. R. 2001 Gradient enhancement and filament ejection for a non-uniform elliptic vortex in two-dimensional turbulence. J. Fluid Mech. 439, 43–56.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.Google Scholar
Kraichnan, R. H. 1971 Inertial-range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47, 525–535.Google Scholar
Kraichnan, R. H. 1974 Kolmogorovs inertial-range theories. J. Fluid Mech. 62, 305–330.Google Scholar
Kraichnan, R. H. 1976 Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 1521–1536.2.0.CO;2>CrossRefGoogle Scholar
Lapeyre, G., Klein, P. & Hua, B. L. 1999 Does the tracer gradient align with the strain eigenvectors in 2d turbulence? Phys. Fluids 11, 3729–3737.CrossRefGoogle Scholar
Lilly, D. K. 1969 Numerical simulation of two-dimensional turbulence. Phys. Fluids Suppl. 12 (12), 240–249.Google Scholar
Lilly, D. K. 1972 Numerical simulation studies of two-dimensional turbulence: I. Models of statistically steady turbulence. Geophys. Fluid Dyn. 3, 289–319.CrossRefGoogle Scholar
Lilly, D. K. 1989 Two-dimensional turbulence generated by energy sources at two scales. J. Atmos. Sci. 46 (13), 2026–2030.Google Scholar
Maltrud, M. E. & Vallis, G. K. 1991 Energy-spectra and coherent structures in forced 2-dimensional and beta-plane turbulence. J. Fluid Mech. 228, 321–342.Google Scholar
Maltrud, M. E. & Vallis, G. K. 1993 Energy and enstrophy transfer in numerical simulations of two-dimensional turbulence. Phys. Fluids A 5, 1760–1775.Google Scholar
McWilliams, J. C. 1990 The vortices of 2-dimensional turbulence. J. Fluid Mech. 219, 361–385.Google Scholar
Meneveau, C. & Katz, J. 2000 Scale-invariance and turbulence models for large-eddy simulation. Ann. Rev. Fluid Mech. 32, 1–32.Google Scholar
Nadiga, B. 2006 On zonal jets in oceans. Geo. Res. Lett. 33, L10601.Google Scholar
Nielsen, A. H., He, X., Rasmussen, J. J. & Bohr, T. 1996 Vortex merging and spectral cascade in two-dimensional flows. Phys. Fluids 8, 2263–2265.Google Scholar
Okhitani, K. & Kida, S. 1992 Triad interactions in a forced turbulence. Phys. Fluids A 4, 794–802.Google Scholar
Okubo, A. 1970 Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences. Deep-Sea Res. 17, 445–454.Google Scholar
Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento Suppl. 6, 279–287.Google Scholar
Paret, J. & Tabeling, P. 1998 Intermitency in the two-dimensional inverse cascade of energy: experimental observations. Phys. Fluids 10, 3126–3136.Google Scholar
Read, P. L. 2001 Transition to geostrophic turbulence in the laboratory, and as a paradigm in atmospheres and oceans. Surv. Geophys. 22 (3), 265–317.CrossRefGoogle Scholar
Rhines, P. B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–443.Google Scholar
Rhines, P. B. 1979 Geostrophic turbulence. Ann. Rev. Fluid Mech. 11, 401–441.Google Scholar
Rivera, M. K. 2000 The inverse energy cascade of two-dimensional turbulence. PhD thesis, University of Pittsburgh, Pittsburgh.Google Scholar
Salmon, R. 1982 Geostrophic turbulence. In Topics in Ocean Physics, Proc. International School of Physics “Enrico Fermi” (ed. Osborne, A. R. & Rizzoli, P. M.), pp. 30–78. North Holland.Google Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.Google Scholar
Scott, R. K. 2007 Nonrobustness of the two-dimensional turbulent inverse cascade. Phys. Rev. E 75 (4), 046301.Google Scholar
Scott, R. B. & Arbic, B. K. 2007 Spectral energy fluxes in geostrophic turbulence: implications for ocean energetics. J. Phys. Oceanogr. 37 (3), 673–688.Google Scholar
Scott, R. K. & Polvani, L. M. 2007 Forced-dissipative shallow-water turbulence on the sphere and the atmospheric circulation of the giant planets. J. Atmos. Sci. 64 (9), 3158–3176.Google Scholar
Smith, L. M. & Yakhot, V. 1993 Bose condensation and small-scale structure generation in a random force driven 2d turbulence. Phys. Rev. Lett. 71 (3), 352–355.Google Scholar
Starr, V. P. 1953 Note concerning the nature of the large-scale eddies in the atmosphere. Tellus 5, 494–498.Google Scholar
Starr, V. P. 1968 Physics of Negative Viscosity Phenomena. McGraw-Hill.Google Scholar
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry, and Engineering. Perseus Publishing.Google Scholar
Sukoriansky, S., Chekhlov, A., Galperin, B. & Orszag, S. A. 1996 Large eddy simulation of two-dimensional isotropic turbulence. J. Sci. Comp. 11 (1), 13–45.Google Scholar
Sukoriansky, S., Dikovskaya, N. & Galperin, B. 2007 On the “arrest” of inverse energy cascade and the rhines scale. J. Atmos. Sci. 64 (9), 3312–3327.Google Scholar
Sukoriansky, S., Galperin, B. & Chekhlov, A. 1999 Large scale drag representation in simulations of two-dimensional turbulence. Phys. Fluids 11 (10), 3043–3053.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.Google Scholar
Tran, C. V. 2006 Diminishing inverse transfer and non-cascading dynamics in surface quasi-geostrophic turbulence. Phys. D 213 (1), 76–84.Google Scholar
Tran, C. V. & Bowman, J. C. 2003 On the dual cascade in two-dimensional turbulence. Phys. D 176 (3–4), 242–255.Google Scholar
Tran, C. V. & Bowman, J. C. 2004 Robustness of the inverse cascade in two-dimensional turbulence. Phys. Rev. E 69 (3), 036303.Google Scholar
Tulloch, R. & Smith, K. S. 2006 A theory for the atmospheric energy spectrum: depth-limited temperature anomalies at the tropopause. Proc. Natl. Acad. Sci. U. S. A. 103 (40), 14690–14694.Google Scholar
Tung, K. K. & Orlando, W. W. 2003 The k −3 and k −5/3 energy spectrum of atmospheric turbulence: quasigeostrophic two-level model simulation. J. Atmos. Sci. 60 (6), 824–835.Google Scholar
Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A 4, 350–363.Google Scholar
Weiss, J. B. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Phys. D 48 (2–3), 273–294.Google Scholar
Weiss, J. B. & McWilliams, J. C. 1993 Temporal scaling behavior of decaying two-dimensional turbulence. Phys. Fluids A 5 (3), 608–621.Google Scholar
Yeung, P. K. & Brasseur, J. G. 1991 The response of isotropic turbulence to isotropic and anisotropic forcing at the large scales. Phys. Fluids A 3, 884–897.Google Scholar
Zhou, Y. 1993 Interacting scales and energy transfer in turbulence. Phys. Fluids A 5, 2511–2524.Google Scholar