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The turbulent cascade of inertia-gravity waves in rotating shallow water

Published online by Cambridge University Press:  25 November 2024

Jim Thomas Open the ORCID record for Jim Thomas [Opens in a new window] ,
Rajendra S. Rajpoot Open the ORCID record for Rajendra S. Rajpoot [Opens in a new window]  and
Prateek Gupta Open the ORCID record for Prateek Gupta [Opens in a new window]
Jim Thomas*
Affiliation:
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore 560065, India
Rajendra S. Rajpoot
Affiliation:
Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore 560065, India
Prateek Gupta
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi 110016, India
*
†Email address for correspondence: jimthomas.edu@gmail.com
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Abstract

In this work we study features of inertia-gravity wave turbulence in the rotating shallow water equations. On examining the dynamics of waves with varying rotation rates, we find that the turbulent cascade of waves is strongest at low rotation rates, forming a $k^{-2}$ energy spectrum, and a rich distribution of shocks in physical space. At high rotation rates, the forward cascade of waves weakens along with a steeper energy spectra and vanishing of shocks in physical space. The wave cascade is seen to be scale-local, resulting in a noticeable time interval for energy to get transferred from domain scale to dissipative scale. Furthermore, we find that the vortical flow has a non-negligible effect on the wave cascade, especially at high rotation rates. The vortical flow assists in the forward cascade of waves and shock formation at high rotation rates, while the waves by themselves in the absence of the vortical flow lack a forward cascade and shock formation at such high rotation rates. On investigating the physical space structures in the vortical flow and their connections to the wave cascade, we find that strain-dominant regions, that are located around the boundaries of coherent vortices, are the physical space regions that contribute majorly to the forward cascade of waves. Our results in general highlight intriguing features of dispersive inertia-gravity wave turbulence that are qualitatively similar to those seen in three-dimensional homogeneous isotropic turbulence and are beyond the predictions of asymptotic resonant wave interaction theory.

JFM classification

Turbulent Flows: Wave-turbulence interactions Geophysical and Geological Flows: Waves in rotating fluids Geophysical and Geological Flows: Shallow water flows

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Type
JFM Papers
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Journal of Fluid Mechanics , Volume 1000 , 10 December 2024 , A30
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© Jim Thomas, 2024. Published by Cambridge University Press

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References

Arbic, B.K., Scott, R.B., Flierl, G.R., Morten, A.J., Richman, J.G. & Shriver, J.F. 2012 Nonlinear cascades of surface oceanic geostrophic kinetic energy in the frequency domain. J. Phys. Oceanogr. 42, 1577–1600.CrossRefGoogle Scholar
Augier, P., Mohanan, A.V. & Lindborg, E. 2019 Shallow water wave turbulence. J. Fluid Mech. 874, 1169–1196.CrossRefGoogle Scholar
Babin, A., Mahalov, A. & Nicolaenko, B. 1997 Global splitting and regularity of rotating shallow-water equations. Eur. J. Mech. (B/Fluids) 16 (1), 725–754.Google Scholar
Banerjee, D. 2019 Fractional hyperviscosity induced growth of bottlenecks in energy spectrum of Burgers equation solutions. Eur. Phys. J. B 92, 209.CrossRefGoogle Scholar
Bartello, P. 1995 Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atmos. Sci. 52, 4410–4428.2.0.CO;2>CrossRefGoogle Scholar
Brasseur, J.G. & Wei, C.-H. 1994 Interscale dynamics and local isotropy in high Reynolds number turbulence within triadic interaction. Phys. Fluids 6, 842–870.CrossRefGoogle Scholar
Cardesa, J.I., Vela-Martin, A., Dong, S. & Jimenez, J. 2015 The temporal evolution of the energy flux across scales in homogeneous turbulence. Phys. Fluids 27, 111702.CrossRefGoogle Scholar
Cardesa, J.I., Vela-Martin, A. & Jimenez, J. 2017 The turbulent cascade in five dimensions. Science 357, 782–784.CrossRefGoogle ScholarPubMed
Craik, A.D.D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.Google Scholar
Domaradzki, J.A. & Carati, D. 2007 An analysis of the energy transfer and the locality of nonlinear interactions in turbulence. Phys. Fluids 19, 085112.CrossRefGoogle Scholar
Domaradzki, J.A. & Rogallo, R.S. 1990 Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence. Phys. Fluids A 2, 413–426.CrossRefGoogle Scholar
Eyink, G.L. & Aluie, H. 2009 Localness of energy cascade in hydrodynamic turbulence. I. Smooth coarse graining. Phys. Fluids 21, 115107.CrossRefGoogle Scholar
Falkovich, G. & Kritsuk, A.G. 2017 How vortices and shocks provide for a flux loop in two-dimensional compressible turbulence. Phys. Rev. Fluids 2R, 092603.CrossRefGoogle Scholar
Farge, M. & Sadourny, R. 1989 Wave-vortex dynamics in rotating shallow water. J. Fluid Mech. 206, 433–462.CrossRefGoogle Scholar
Frisch, U., Ray, S.S., Sahoo, G., Banerjee, D. & Pandit, R. 2013 Real-space manifestations of bottlenecks in turbulence spectra. Phys. Rev. Lett. 110, 064501.CrossRefGoogle ScholarPubMed
Gertz, A. & Straub, D.N. 2009 Near-inertial oscillations and the damping of midlatitude gyres: a modelling study. J. Phys. Oceanogr. 39, 2338–2350.CrossRefGoogle Scholar
Jossy, J.P. & Gupta, P. 2023 Baroclinic interaction of forced shock waves with random thermal gradients. Phys. Fluids 35, 056114.CrossRefGoogle Scholar
Kafiabad, H.A., Savva, M.A. & Vanneste, J. 2019 Diffusion of inertia-gravity waves by geostrophic turbulence. J. Fluid Mech. 869, R7.CrossRefGoogle Scholar
Khurshid, S., Donzis, D.A. & Sreenivasan, K.R. 2021 Slow spectral transfer and energy cascades in isotropic turbulence. J. Fluid Mech. 908, A21.CrossRefGoogle Scholar
Kuznetsov, E. 2004 Turbulence spectra generated by singularities. JETP Lett. 80, 83–89.CrossRefGoogle Scholar
Lahaye, N. & Zeitlin, V. 2012 Decaying vortex and wave turbulence in rotating shallow water model, as follows from high-resolution direct numerical simulations. Phys. of Fluids 24, 115106.CrossRefGoogle Scholar
Lelong, M.P. & Riley, J.J. 1991 Internal wave-vortical mode interactions in strongly stratified flows. J. Fluid Mech. 232, 1–19.CrossRefGoogle Scholar
Lesieur, M. 2008 Turbulence in Fluids. Fluid Mechanics and its Applications, vol. 84, 4th edn. Springer.Google Scholar
Majda, A.J. 2002 Introduction to Partial Differential Equations and Waves for the Atmosphere and Ocean-Courant Lecture Notes, Bd. 9. American Mathematical Society.Google Scholar
Majda, A.J. & Embid, P. 1998 Averaging over fast gravity waves for geophysical flows with unbalanced initial data. Theor. Comput. Fluid Dyn. 11, 155–169.CrossRefGoogle Scholar
McComb, W.D. 2014 Homogeneous, Isotropic Turbulence: Phenomenology, Renormalization and Statistical Closures. Oxford University Press.CrossRefGoogle Scholar
Muller, M., Arbic, B.K., Richman, J.G., Shriver, J.F., Kunze, E.L., Scott, R.B., Wallcraft, A.J. & Zamudio, L. 2015 Toward an internal gravity wave spectrum in global ocean models. Geophys. Res. Lett. 42, 3474–3481.CrossRefGoogle Scholar
Ohkitani, K. & Kida, S. 1992 Triad interactions in a forced turbulence. Phys. Fluids A4, 794–802.CrossRefGoogle 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
Pearson, B., Fox-Kemper, B., Bachman, S. & Bryan, F. 2017 Evaluation of scale-aware subgrid mesoscale eddy models in a global eddy-rich model. Ocean Model. 115, 42–58.CrossRefGoogle Scholar
Phillips, O.M. 1981 Wave interactions – the evolution of an idea. J. Fluid Mech. 106, 215–227.CrossRefGoogle Scholar
Polvani, L.M., McWilliams, J.C., Spall, M.A. & Ford, R. 1994 The coherent structures of shallow-water turbulence: deformation-radius effects, cyclone/anticyclone asymmetry and gravity-wave generation. Chaos 4, 177–186.CrossRefGoogle ScholarPubMed
Remmel, M. & Smith, L. 2009 New intermediate models for rotating shallow water and an investigation of the preference for anticyclones. J. Fluid Mech. 635, 321–359.CrossRefGoogle Scholar
Rocha, C.B., Wagner, G.L. & Young, W.R. 2018 Stimulated generation-extraction of energy from balanced flow by near-inertial waves. J. Fluid Mech. 847, 417–451.CrossRefGoogle Scholar
Salmon, R. 1978 Lectures on Geophysical Fluid Dynamics. Oxford University Press.Google Scholar
Savva, M.A., Kafiabad, H.A. & Vanneste, J. 2021 Inertia-gravity-wave scattering by three-dimensional geostrophic turbulence. J. Fluid Mech. 916, A6.CrossRefGoogle Scholar
Taylor, S. & Straub, D. 2016 Forced near-inertial motion and dissipation of low-frequency kinetic energy in a wind-driven channel flow. J. Phys. Oceanogr. 46, 79–93.CrossRefGoogle Scholar
Taylor, S. & Straub, D. 2020 Effects of adding forced near-inertial motion to a wind-driven channel flow. J. Phys. Oceanogr. 50, 2983–2996.CrossRefGoogle Scholar
Thomas, J. 2023 Turbulent wave-balance exchanges in the ocean. Proc. R. Soc. Lond. A 479, 20220565.Google Scholar
Thomas, J. & Arun, S. 2020 Near-inertial waves and geostrophic turbulence. Phys. Rev. Fluids 5, 014801.CrossRefGoogle Scholar
Thomas, J. & Daniel, D. 2020 Turbulent exchanges between near-inertial waves and balanced flows. J. Fluid Mech. 902, A7.CrossRefGoogle Scholar
Thomas, J. & Daniel, D. 2021 Forward flux and enhanced dissipation of geostrophic balanced energy. J. Fluid Mech. 911, A60.CrossRefGoogle Scholar
Thomas, J. & Ding, L. 2023 Upscale transfer of waves in one-dimensional rotating shallow water. J. Fluid Mech. 961, A2.CrossRefGoogle Scholar
Thomas, J. & Yamada, R. 2019 Geophysical turbulence dominated by inertia-gravity waves. J. Fluid Mech. 875, 71–100.CrossRefGoogle Scholar
Vallis, G.K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Wagner, G.L. & Young, W.R. 2016 A three-component model for the coupled evolution of near-inertial waves, quasi-geostrophic flow and the near-inertial second harmonic. J. Fluid Mech. 802, 806–837.CrossRefGoogle Scholar
Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A4, 350–363.CrossRefGoogle Scholar
Ward, M.L. & Dewar, W.K. 2010 Scattering of gravity waves by potential vorticity in a shallow-water fluid. J. Fluid Mech. 663, 478–506.CrossRefGoogle Scholar
Warn, T. 1986 Statistical mechanical equilibria of the shallow water equations. Tellus A 38A, 1–11.CrossRefGoogle Scholar
Weiss, J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273–294.CrossRefGoogle Scholar
Xie, J.H. & Vanneste, J. 2015 A generalised-Lagrangian-mean model of the interactions between near-inertial waves and mean flow. J. Fluid Mech. 774, 143–169.CrossRefGoogle Scholar
Xie, J.-H. 2020 Downscale transfer of quasigeostrophic energy catalyzed by near-inertial waves. J. Fluid Mech. 904, A40.CrossRefGoogle Scholar
Yeung, P.K. & Brasseur, J.G. 1991 The response of isotropic turbulence to isotropic and anisotropic forcing at the large scales. Phys. Fluids A3, 884–897.CrossRefGoogle Scholar
Yuan, L. & Hamilton, K. 1994 Equilibrium dynamics in a forced-dissipative f-plane shallow-water system. J. Fluid Mech. 280, 369–394.CrossRefGoogle Scholar
Zeitlin, V. 2018 Geophysical Fluid Dynamics: Understanding (Almost) Everything with Rotating Shallow Water Models. Oxford University Press.CrossRefGoogle Scholar
Zhang, L.F. & Xie, J.H. 2023 The catalytic effect of near-inertial waves on beta-plane zonal jets. J. Fluid Mech. 962, A33.CrossRefGoogle Scholar
Zhou, Y. 1993 Degrees of locality of energy transfer in the inertial range. Phys. Fluids A 5, 1092–1094.CrossRefGoogle Scholar