Hostname: page-component-5cf477f64f-h6p2m Total loading time: 0 Render date: 2025-04-03T00:17:44.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Cascades transition in generalised two-dimensional turbulence

Published online by Cambridge University Press:  02 April 2025

Vibhuti Bhushan Jha Open the ORCID record for Vibhuti Bhushan Jha [Opens in a new window] ,
Kannabiran Seshasayanan Open the ORCID record for Kannabiran Seshasayanan [Opens in a new window]  and
Vassilios Dallas Open the ORCID record for Vassilios Dallas [Opens in a new window]
Vibhuti Bhushan Jha*
Affiliation:
Space Applications Centre, Indian Space Research Organisation, Ahmedabad, Gujarat, India Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, India
Kannabiran Seshasayanan*
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, India
Vassilios Dallas*
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK Environmental Research Laboratory, National Centre for Scientific Research “Demokritos”, 15341 Athens, Greece
*
Corresponding authors: Vibhuti Bhushan Jha, [email protected]; Kannabiran Seshasayanan, [email protected]; Vassilios Dallas, [email protected]
Corresponding authors: Vibhuti Bhushan Jha, [email protected]; Kannabiran Seshasayanan, [email protected]; Vassilios Dallas, [email protected]
Corresponding authors: Vibhuti Bhushan Jha, [email protected]; Kannabiran Seshasayanan, [email protected]; Vassilios Dallas, [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Generalised two-dimensional (2-D) fluid dynamics is characterised by a relationship between a scalar field $q$, called generalised vorticity, and the stream function $\psi$,namely $q = (-\nabla ^2)^{\frac {\alpha }{2}} \psi$. We study the transition of cascades in generalised 2-D turbulence by systematically varying the parameter $\alpha$ and investigating its influential role in determining the directionality (inverse, forward or bidirectional) of these cascades. We derive upper bounds for the dimensionless dissipation rates of generalised energy $E_G$ and enstrophy $\Omega _G$ as the Reynolds number tends to infinity. These findings corroborate numerical simulations, illustrating the inverse cascade of $E_G$ and forward cascade of $\Omega _G$ for $\alpha \gt 0$, contrasting with the reverse behaviour for $\alpha \lt 0$. The dependence of dissipation rates on system parameters reinforces these observed transitions, substantiated by spectral fluxes and energy spectra, which hint at Kolmogorov-like scalings at large scales but discrepancies at smaller scales between numerical and theoretical estimates. These discrepancies are possibly due to non-local transfers, which dominate the dynamics as we go from positive to negative values of $\alpha$. Intriguingly, the forward cascade of $E_G$ for $\alpha \lt 0$ reveals similarities to three-dimensional turbulence, notably the emergence of vortex filaments within a 2-D framework, marking a unique feature of this generalised model.

JFM classification

Geophysical and Geological Flows: Quasi-geostrophic flows Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1008 , 10 April 2025 , A23
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

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

Alexakis, A. 2011 Two-dimensional behavior of three-dimensional magnetohydrodynamic flow with a strong guiding field. Phys. Rev. E 84 (5), 056330.Google Scholar
Alexakis, A. 2018 Three-dimensional instabilities and negative eddy viscosity in thin-layer flows. Phys. Rev. Fluids 3 (11), 114601.CrossRefGoogle Scholar
Alexakis, A. 2023 Quasi-two-dimensional turbulence. Rev. Mod. Plasma Phys. 7 (1), 31.CrossRefGoogle Scholar
Alexakis, A. & Biferale, L. 2018 Cascades and transitions in turbulent flows. Phys. Rep. 767–769, 1–101.CrossRefGoogle Scholar
Alexakis, A. & Biferale, L. 2022 λ-Navier–Stokes turbulence. Phil. Trans. R. Soc. Lond. A: Math. Phys. Engng Sci. 380, 20210243.Google ScholarPubMed
Alexakis, A. & Doering, C.R. 2006 Energy and enstrophy dissipation in steady state 2d turbulence. Phys. Lett. A 359 (6), 652–657.CrossRefGoogle Scholar
Arbic, B.K., Polzin, K.L., Scott, R.B., Richman, J.G. & Shriver, J.F. 2013 On eddy viscosity, energy cascades, and the horizontal resolution of gridded satellite altimeter products. J. Phys. Oceanogr. 43 (2), 283–300.CrossRefGoogle Scholar
Balwada, D., LaCasce, J.H. & Speer, K.G. 2016 Scale‐dependent distribution of kinetic energy from surface drifters in the Gulf of Mexico. Geophys. Res. Lett. 43 (20), 10–856.CrossRefGoogle Scholar
Balwada, D., Xie, J.-H., Marino, R. & Feraco, F. 2022 Direct observational evidence of an oceanic dual kinetic energy cascade and its seasonality. Sci. Adv. 8 (41), eabq2566.CrossRefGoogle ScholarPubMed
Benavides, S.J. & Alexakis, A. 2017 Critical transitions in thin layer turbulence. J. Fluid Mech. 822, 364–385.Google Scholar
Boffetta, G. & Ecke, R.E. 2012 Two-dimensional turbulence. Annu. Rev. Fluid Mech. 44 (1), 427–451.CrossRefGoogle Scholar
Boffetta, G. & Musacchio, S. 2010 Evidence for the double cascade scenario in two-dimensional turbulence. Phys. Rev. E 82 (1), 016307.Google ScholarPubMed
Byrne, D. & Zhang, J.A. 2013 Height‐dependent transition from 3‐D to 2‐D turbulence in the hurricane boundary layer. Geophys. Res. Lett. 40 (7), 1439–1442.Google Scholar
Campagne, A., Gallet, B., Moisy, F. & Cortet, P.-P. 2014 Direct and inverse energy cascades in a forced rotating turbulence experiment. Phys. Fluids 26 (12), 125112.CrossRefGoogle Scholar
Carton, X., Ciani, D., Verron, J., Reinaud, J. & Sokolovskiy, M. 2016 Vortex merger in surface quasi-geostrophy. Geophys. Astrophys. Fluid Dyn. 110 (1), 1–22.CrossRefGoogle Scholar
Celani, A., Musacchio, S. & Vincenzi, D. 2010 Turbulence in more than two and less than three dimensions. Phys. Rev. Lett. 104 (18), 184506.Google ScholarPubMed
Chan, C.-k., Mitra, D. & Brandenburg, A. 2012 Dynamics of saturated energy condensation in two-dimensional turbulence. Phys. Rev. E 85 (3), 036315.Google ScholarPubMed
Deusebio, E., Boffetta, G., Lindborg, E. & Musacchio, S. 2014 Dimensional transition in rotating turbulence. Phys. Rev. E 90 (2), 023005.Google ScholarPubMed
Doering, C.R. & Gibbon, J.D. 1995 Applied Analysis of the Navier–Stokes Equations. Cambridge University Press.Google Scholar
Ecke, R.E. 2017 From 2D to 3D in fluid turbulence: unexpected critical transitions. J. Fluid Mech. 828, 1–4.CrossRefGoogle Scholar
Fjørtoft, R. 1953 On the changes in the spectral distribution of kinetic energy for twodimensional, nondivergent flow. Tellus A: Dyn. Meteorol. Oceanogr. 5 (3), 225–230.CrossRefGoogle Scholar
Foussard, A., Berti, S., Perrot, X. & Lapeyre, G. 2017 Relative dispersion in generalized two-dimensional turbulence. J. Fluid Mech. 821, 358–383.Google Scholar
Frisch, U., Lesieur, M. & Sulem, P.L. 1976 Crossover dimensions for fully developed turbulence. Phys. Rev. Lett. 37 (14), 895–897.CrossRefGoogle Scholar
Gottlieb, D. & Orszag, S.A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. SIAM. Society for Industrial and Applied Mathematics Philadelphia.CrossRefGoogle Scholar
Held, I.M., Pierrehumbert, R.T., Garner, S.T. & Swanson, K.L. 1995 a Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20.Google Scholar
Held, I.M., Pierrehumbert, R.T., Garner, S.T. & Swanson, K.L. 1995 b Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20.Google Scholar
Ishihara, T., Gotoh, T. & Kaneda, Y. 2009 Study of high–reynolds number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41 (1), 165–180.CrossRefGoogle Scholar
Iwayama, T., Murakami, S. & Watanabe, T. 2015 Anomalous eddy viscosity for two-dimensional turbulence. Phys. Fluids. 27 (4), 045104.Google Scholar
Iwayama, T. & Watanabe, T. 2010 Green’s function for a generalized two-dimensional fluid. Phys. Rev. E 82 (3), 036307.Google ScholarPubMed
van Kan, A. 2024 Phase transitions in anisotropic turbulence. Chaos 34 (12), 122103.Google ScholarPubMed
van Kan, A. & Alexakis, A. 2020 Critical transition in fast-rotating turbulence within highly elongated domains. J. Fluid Mech. 899, A33.Google Scholar
Khatri, H., Sukhatme, J., Kumar, A. & Verma, M.K. 2018 Surface ocean enstrophy, kinetic energy fluxes, and spectra from satellite altimetry. J. Geophys. Res.: Oceans 123 (5), 3875–3892.Google Scholar
King, G.P., Vogelzang, J. & Stoffelen, A. 2015 Upscale and downscale energy transfer over the tropical Pacific revealed by scatterometer winds. J. Geophys. Res.: Oceans 120 (1), 346–361.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Dokl. Akad. Nauk SSSR 30, 301.Google Scholar
Kraichnan, R.H. 1971 Inertial-range transfer in two-and three-dimensional turbulence. J. Fluid Mech. 47 (3), 525–535.CrossRefGoogle Scholar
Kraichnan, R.H. 1976 Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33 (8), 1521–1536.Google Scholar
Lapeyre, G. 2017 Surface quasi-geostrophy. Fluids 2 (1), 7.CrossRefGoogle Scholar
Larichev, V.D. & McWilliams, J.C. 1991 Weakly decaying turbulence in an equivalent-barotropic fluid. Phys. Fluids A 3 (5), 938–950.Google Scholar
Lesur, G. & Longaretti, P.-Y. 2011 Non-linear energy transfers in accretion discs MRI turbulence. Astron. Astrophys. 528, A17.CrossRefGoogle Scholar
Lilly, D.K. 1969 Numerical simulation of two-dimensional turbulence. Phys. Fluids 12 (12), II–240.Google Scholar
Marino, R., Mininni, P.D., Rosenberg, D. & Pouquet, A. 2013 Inverse cascades in rotating stratified turbulence: fast growth of large scales. Europhys. Lett. 102 (4), 44006.CrossRefGoogle Scholar
Nastrom, G., Gage, K. & Jasperson, W. 1984 Kinetic energy spectrum of large-and mesoscale atmospheric processes. Nature 310 (5972), 36–38.Google Scholar
Pierrehumbert, R.T., Held, I.M. & Swanson, K.L. 1994 a Spectra of local and nonlocal two-dimensional turbulence. Chaos, Solitons Fractals 4 (6), 1111–1116.CrossRefGoogle Scholar
Pierrehumbert, R.T., Held, I.M. & Swanson, K.L. 1994 b Spectra of local and nonlocal two-dimensional turbulence. Chaos, Solitons Fractals 4 (6), 1111–1116.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. 1979 Geostrophic turbulence. Annu. Rev. Fluid Mech. 11 (1), 401–441.CrossRefGoogle Scholar
Scott, R.B. & Wang, F. 2005 Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry. J. Phys. Oceanogr. 35 (9), 1650–1666.Google Scholar
Scott, R. K. & Dritschel, D. 2014 Numerical simulation of a self-similar cascade of filament instabilities in the surface quasigeostrophic system. Phys. Rev. Lett. 112 (14), 144505.Google ScholarPubMed
Sen, A., Mininni, P.D., Rosenberg, D. & Pouquet, A. 2012 Anisotropy and nonuniversality in scaling laws of the large-scale energy spectrum in rotating turbulence. Phys. Rev. E 86 (3), 036319.Google ScholarPubMed
Seshasayanan, K., Benavides, S.J. & Alexakis, A. 2014 a On the edge of an inverse cascade. Phys. Rev. E 90 (5), 051003.Google ScholarPubMed
Seshasayanan, K., Benavides, S.J. & Alexakis, A. 2014 b On the edge of an inverse cascade. Phys. Rev. E 90 (5), 051003.Google ScholarPubMed
Shao, X., Zhang, N. & Tang, J. 2023 A physical model for the observed inverse energy cascade in typhoon boundary layers. Geophys. Res. Lett. 50 (18), 264501.Google Scholar
Shats, M., Byrne, D. & Xia, H. 2010 Turbulence decay rate as a measure of flow dimensionality. Phys. Rev. Lett. 105 (26), 264501.Google ScholarPubMed
Siegelman, L., et al. 2022 Moist convection drives an upscale energy transfer at Jovian high latitudes. Nat. Phys. 18 (3), 357–361.Google Scholar
Sivashinsky, G. & Yakhot, V. 1985 Negative viscosity effect in large-scale flows. Phys. Fluids 28 (4), 1040–1042.Google Scholar
Smith, K., Boccaletti, G., Henning, C., Marinov, I., Tam, C., Held, I. & Vallis, G. 2002 Turbulent diffusion in the geostrophic inverse cascade. J. Fluid Mech. 469, 13–48.Google Scholar
Smith, L.M., Chasnov, J.R. & Waleffe, F. 1996 Crossover from two-to three-dimensional turbulence. Phys. Rev. Lett. 77 (12), 2467–2470.CrossRefGoogle ScholarPubMed
Sozza, A., Boffetta, G., Muratore-Ginanneschi, P. & Musacchio, S. 2015 Dimensional transition of energy cascades in stably stratified forced thin fluid layers. Phys. Fluids 27 (3), 035112.Google Scholar
Tabeling, P. 2002 Two-dimensional turbulence: a physicist approach. Phys. Rep. 362 (1), 1–62.Google Scholar
Teitelbaum, T. & Mininni, P.D. 2012 Thermalization and free decay in surface quasigeostrophic flows. Phys. Rev. E 86 (1), 016323.Google ScholarPubMed
Tran, C.V. 2004 Nonlinear transfer and spectral distribution of energy in $\alpha$ turbulence. Phys. D: Nonlinear Phenom. 191 (1-2), 137–155.CrossRefGoogle Scholar
Verma, M.K. 2024 Critical dimension for hydrodynamic turbulence. Phys. Rev. E 110 (3), 035102.Google ScholarPubMed
Watanabe, T. & Iwayama, T. 2004 Unified scaling theory for local and non-local transfers in generalized two-dimensional turbulence. J. Phys. Soc. Japan 73 (12), 3319–3330.CrossRefGoogle Scholar
Watanabe, T. & Iwayama, T. 2007 Interacting scales and triad enstrophy transfers in generalized two-dimensional turbulence. Phys. Rev. E 76 (4), 046303.Google ScholarPubMed
Xia, H., Byrne, D., Falkovich, G. & Shats, M. 2011 Upscale energy transfer in thick turbulent fluid layers. Nat. Phys. 7 (4), 321–324.CrossRefGoogle Scholar
Young, R.M. & Read, P.L. 2017 Forward and inverse kinetic energy cascades in Jupiter’s turbulent weather layer. Nat. Phys. 13 (11), 1135–1140.CrossRefGoogle Scholar