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Two-dimensional isotropic inertia–gravity wave turbulence

Published online by Cambridge University Press:  14 June 2019

Jin-Han Xie Open the ORCID record for Jin-Han Xie [Opens in a new window]  and
Oliver Bühler Open the ORCID record for Oliver Bühler [Opens in a new window]
Jin-Han Xie*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Oliver Bühler
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
*
†Email address for correspondence: jhxie@cims.nyu.edu
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Abstract

We present an idealized study of rotating stratified wave turbulence in a two-dimensional vertical slice model of the Boussinesq equations, focusing on the peculiar case of equal Coriolis and buoyancy frequencies. In this case the fully nonlinear fluid dynamics can be shown to be isotropic in the vertical plane, which allows the classical methods of isotropic turbulence to be applied. Contrary to ordinary two-dimensional turbulence, here a robust downscale flux of total energy is observed in numerical simulations that span the full parameter regime between Ozmidov and forcing scales. Notably, this robust downscale flux of the total energy does not hold separately for its various kinetic and potential components, which can exhibit both upscale and downscale fluxes, depending on the parameter regime. Using a suitable extension of the classical Kármán–Howarth–Monin equation, exact expressions that link third-order structure functions and the spectral energy flux are derived and tested against numerical results. These expressions make obvious that even though the total energy is robustly transferred downscale, the third-order structure functions are sign indefinite, which illustrates that the sign and the form of measured third-order structure functions are both crucially important in determining the direction of the spectral energy transfer.

JFM classification

Turbulent Flows: Isotropic turbulence Turbulent Flows: Rotating turbulence Turbulent Flows: Stratified turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 872 , 10 August 2019 , pp. 752 - 783
Copyright
© 2019 Cambridge University Press 

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References

Alexakis, A. & Biferale, L. 2018 Cascades and transitions in turbulent flows. Phys. Rep. 767–769, 1–101.Google Scholar
Augier, P., Galtier, S. & Billant, P. 2012 Kolmogorov laws for stratified turbulence. J. Fluid Mech. 709, 659–670.Google Scholar
Bartello, P. 1995 Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atmos. Sci. 52 (24), 4410–4428.Google Scholar
Bernard, D. 1999 Three-point velocity correlation functions in two-dimensional forced turbulence. Phys. Rev. E 60 (5), 6184–6187.Google Scholar
Biferale, L., Bonaccorso, F., Mazzitelli, I. M., van Hinsberg, M. A. T., Lanotte, A. S., Musacchio, S., Perlekar, P. & Toschi, F. 2016 Coherent structures and extreme events in rotating multiphase turbulent flows. Phys. Rev. X 6, 041036.Google Scholar
Bühler, O., Callies, J. & Ferrari, R. 2014 Wave-vortex decomposition of one-dimensional ship-track data. J. Fluid Mech. 756, 1007–1026.Google Scholar
Bühler, O. & McIntyre, M. E. 2005 Wave capture and wavevortex duality. J. Fluid Mech. 534, 67–95.Google Scholar
Callies, J., Bühler, O. & Ferrari, R. 2016 The dynamics of mesoscale winds in the upper troposphere and lower stratosphere. J. Atmos. Sci. 73 (12), 4853–4872.Google Scholar
Callies, J., Ferrari, R. & Bühler, O. 2014 Transition from geostrophic turbulence to inertia-gravity waves in the atmospheric energy spectrum. Proc. Natl Acad. Sci. 111.48, 17033–17038.Google Scholar
Cerbus, R. T. & Chakraborty, P. 2017 The third-order structure function in two dimensions: the Rashomon effect. Phys. Fluids 29, 111110.Google Scholar
Charney, J. G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28, 1087–1095.Google Scholar
Chkhetiani, O. G. 1996 On the third moments in helical turbulence. J. Expl Theor. Phys. 63, 768.Google Scholar
Cho, J. Y. N. & 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
Deusebio, E., Augier, P. & Lindborg, E. 2014a Third-order structure functions in rotating and stratified turbulence: a comparison between numerical, analytical and observational results. J. Fluid Mech. 755, 294–313.Google Scholar
Deusebio, E., Boffetta, G., Lindborg, E. & Musacchio, S. 2014b Dimensional transition in rotating turbulence. Phys. Rev. E 90, 023005.Google Scholar
Deusebio, E., Vallgren, A. & Lindborg, E. 2013 The route to dissipation in strongly stratified and rotating flows. J. Fluid Mech. 720, 66–103.Google Scholar
Dewan, E. M. 1979 Stratospheric wave spectra resembling turbulence. Science 204 (4395), 832–835.Google Scholar
Dritschel, D. G. & McKiver, W. J. 2015 Effect of Prandtl’s ratio on balance in geophysical turbulence. J. Fluid Mech. 777, 569–590.Google Scholar
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.Google Scholar
Gomez, T., Politano, H. & Pouquet, A. 2000 Exact relationship for third-order structure functions in helical flows. Phys. Rev. E 61 (5), 5321–5325.Google Scholar
Grossmann, S. & Mertens, P. 1992 Structure functions in two-dimensional turbulence. Z. Phys. B 88, 105–116.Google Scholar
Hernandez-Duenas, G., Smith, L. M. & Stechmann, S. N. 2014 Investigation of Boussinesq dynamics using intermediate models based on wave–vortical interactions. J. Fluid Mech. 747, 247–287.Google Scholar
Kida, S. 1985 Numerical simulations of two-dimensional turbulence with high-symmetry. J. Phys. Soc. Japan 54, 2840.Google Scholar
Knobloch, E. 1982 Nonlinear diffusive instabilities in differentially rotating stars. Geophys. Astrophys. Fluid Dyn. 22 (1–2), 133–158.Google Scholar
Kolmogorov, A. N. 1941 Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18.Google Scholar
Kraichnan, R. H. 1982 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417.Google Scholar
Kurien, S. 2003 The reflection-antisymmetric counterpart of the Kármán–Howarth dynamical equation. Physica D 175, 167–176.Google Scholar
Kurien, S., Smith, L. & Wingate, B. 2006 On the two-point correlation of potential vorticity in rotating and stratified turbulence. J. Fluid Mech. 555, 131–140.Google Scholar
Kurien, S., Wingate, B. & Taylor, M. A. 2008 Anisotropic constraints on energy distribution in rotating and stratified turbulence. Europhys. Lett. 84, 24003.Google Scholar
Kuznetsov, E. A., Naulin, V., Nielsen, A. H. & Rasmussen, J. J. 2007 Effects of sharp vorticity gradients in two-dimensional hydrodynamic turbulence. Phys. Fluids 19, 105110.Google Scholar
Lilly, D. K. 1989 Two-dimensional turbulence generated by energy sources at two scales. J. Atmos. Sci. 46 (13), 2026–2030.Google Scholar
Lindborg, E. 1999 Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J. Fluid Mech. 388, 259–288.Google Scholar
Lindborg, E. 2005 The effect of rotation on the mesoscale energy cascade in the free atmosphere. Geophys. Res. Lett. 32 (1), L01809.Google Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.Google Scholar
Lindborg, E. 2007 Third-order structure function relations for quasi-geostrophic turbulence. J. Fluid Mech. 572, 255–260.Google Scholar
Lindborg, E. & Cho, J. Y. N. 2000 Determining the cascade of passive scalar variance in the lower stratosphere. Phys. Rev. Lett. 85 (26), 5663–5666.Google Scholar
Lindborg, E. & Cho, J. Y. N. 2001 Horizontal velocity structure functions in the upper troposphere and lower stratosphere 2. Theoretical considerations. J. Geophys. Res. 106 (D10), 10233–10241.Google Scholar
Majda, A. J., McLaughlin, D. W. & Tabak, E. G. 1997 A one-dimensional model for dispersive wave turbulence. J. Nonlinear Sci. 6, 9–44.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, 44006.Google Scholar
Marino, R., Pouquet, A. & Rosenberg, D. 2015a Resolving the paradox of oceanic large-scale balance and small-scale mixing. Phys. Rev. Lett. 114, 114504.Google Scholar
Marino, R., Rosenberg, D., Herbert, C. & Pouquet, A. 2015b Interplay of waves and eddies in rotating stratified turbulence and the link with kinetic-potential energy partition. Europhys. Lett. 112, 49001.Google Scholar
Marino, R., Sorriso-Valvo, L., Carbone, V., Noullez, A., Bruno, R. & Bavassano, B. 2008 Heating the solar wind by a magnetohydrodynamic turbulent energy cascade. Astrophys. J. 677, L71–L74.Google Scholar
Marino, R., Sorriso-Valvo, L., D’Amicis, R., Carbone, V., Bruno, R. & Veltri, P. 2012 On the occurrence of the third-order scaling in high latitude solar wind. Astrophys. J. 750, 41.Google Scholar
Métais, O., Bartello, P., Garnier, E., Riley, J. J. & Lesieur, M. 1996 Inverse cascade in stably stratified rotating turbulence. Dyn. Atmos. Oceans 23, 193–203.Google Scholar
Monin, A. S. & Yaglom, A. M.1975 Statistical Fluid Mechanics, Volume II: Mechanics of Turbulence, reprinted 2007. Dover.Google Scholar
Nastrom, G. D. & Gage, K. S. 1985 A climatology of atmospheric wavenumber spectra of wind and termperature observed by commercial aircraft. J. Atmos. Sci. 42 (9), 950–960.Google Scholar
Nazarenko, S. 2011 Wave Turbulence. Springer Science & Business Media.Google Scholar
Nazarenko, S. V. & Schekochihin, A. A. 2011 Critical balance in magnetohydrodynamic, rotating and stratified turbulence: towards a universal scaling conjecture. J. Fluid Mech. 677, 134–153.Google Scholar
Oks, D., Mininni, P. D., Marino, R. & Pouquet, A. 2017 Inverse cascades and resonant triads in rotating and stratified turbulence. Phys. Rev. Fluids 29, 111109.Google Scholar
Pouquet, A., Marino, R., Mininni, P. D. & Rosenberg, D. 2017 Dual constant-flux energy cascades to both large scales and small scales. Phys. Rev. Fluids 29, 111108.Google Scholar
Salmon, R. 1982 Two-layer quasi-geostrophic turbulence in a simple special case. Geophys. Astrophys. Fluid Dyn. 10 (1), 25–52.Google Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.Google Scholar
Seshasayana, K. & Alexakis, A. 2016 Critical behavior in the inverse to forward energy transition in two-dimensional magnetohydrodynamic flow. Phys. Rev. E 93, 013104.Google Scholar
Seshasayana, K., Benavides, S. J. & Alexakis, A. 2014 On the edge of an inverse cascade. Phys. Rev. E 90, 051003(R).Google Scholar
Smith, L. M. & Waleffe, F. 2002 Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451, 145–168.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. USA 103 (40), 14690–14694.Google Scholar
Tulloch, R. & Smith, K. S. 2009 Quasigeostrophic turbulence with explicit surface dynamics: application to the atmospheric energy spectrum. J. Atmos. Sci. 66 (2), 450–467.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, 824–834.Google Scholar
Vallgren, A. & Lindborg, E. 2010 Charney isotropy and equipartition in quasi-geostrophic turbulence. J. Fluid Mech. 656, 448–457.Google Scholar
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.Google Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.Google Scholar
Xie, J.-H. & Bühler, O. 2018 Exact third-order structure functions for two-dimensional turbulence. J. Fluid Mech. 851, 672–686.Google Scholar
Yakhot, V. 1999 Two-dimensional turbulence in the inverse cascade range. Phys. Rev. E 60 (5), 5544–5551.Google Scholar
Zakharov, V. E., L’vov, V. S. & Falkovich, G. 2012 Kolmogorov Spectra of Turbulence I: Wave Turbulence. Springer Science & Business Media.Google Scholar