Skip to main content Accessibility help

The energy flux spectrum of internal waves generated by turbulent convection

  • Louis-Alexandre Couston (a1), Daniel Lecoanet (a2), Benjamin Favier (a1) and Michael Le Bars (a1)


We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of mixed convective and stably stratified fluids. We demonstrate that in the limit of large Rayleigh number ( $Ra\in [4\times 10^{7},10^{9}]$ ) and large stratification (Brunt–Väisälä frequencies $f_{N}\gg f_{c}$ , where $f_{c}$ is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region as described in Lecoanet & Quataert (Mon. Not. R. Astron. Soc., vol. 430 (3), 2013, pp. 2363–2376). Specifically, we demonstrate that the wave energy flux spectrum scales like $k_{\bot }^{4}\,f^{-13/2}$ for weakly damped waves (with $k_{\bot }$ and $f$ the waves’ horizontal wavenumbers and frequencies, respectively), and that the total wave energy flux decays with $z$ , the distance from the convective region, like $z^{-13/8}$ .


Corresponding author

Email address for correspondence:


Hide All
Alexander, M. J., Geller, M., McLandress, C., Polavarapu, S., Preusse, P., Sassi, F., Sato, K., Eckermann, S., Ern, M., Hertzog, A., Kawatani, Y., Pulido, M., Shaw, T. A., Sigmond, M., Vincent, R. & Watanabe, S. 2010 Recent developments in gravity-wave effects in climate models and the global distribution of gravity-wave momentum flux from observations and models. Q. J. R. Meteorol. Soc. 136 (650), 11031124.
Ansong, J. K. & Sutherland, B. R. 2010 Internal gravity waves generated by convective plumes. J. Fluid Mech. 648, 405.
Bordes, G., Venaille, A., Joubaud, S., Odier, P. & Dauxois, T. 2012 Experimental observation of a strong mean flow induced by internal gravity waves. Phys. Fluids 24 (8), 086602.
van den Bremer, T. S. & Sutherland, B. R. 2018 The wave-induced flow of internal gravity wavepackets with arbitrary aspect ratio. J. Fluid Mech. 834, 385408.
Burns, K. J., Vasil, G. M., Oishi, J. S., Lecoanet, D. & Brown, B. P. 2018 Dedalus: Flexible Framework for Spectrally Solving Differential Equations. Astrophysics Source Code Library.
Canet, L., Rossetto, V., Wschebor, N. & Balarac, G. 2017 Spatiotemporal velocity–velocity correlation function in fully developed turbulence. Phys. Rev. E 95, 023107.
Carruthers, D. J. & Hunt, J. C. R. 1986 Velocity fluctuations near an interface between a turbulent region and a stably stratified layer. J. Fluid Mech. 165, 475501.
Chen, S. & Kraichnan, R. H. 1989 Sweeping decorrelation in isotropic turbulence. Phys. Fluids A 1 (12), 20192024.
Chevillard, L., Roux, S. G., Lévêque, E., Mordant, N., Pinton, J.-F. & Arnéodo, A. 2005 Intermittency of velocity time increments in turbulence. Phys. Rev. Lett. 95, 064501.
Couston, L.-A., Lecoanet, D., Favier, B. & Le Bars, M. 2017 Dynamics of mixed convective–stably-stratified fluids. Phys. Rev. Fluids 2, 094804.
Couston, L.-A., Lecoanet, D., Favier, B. & Le Bars, M. 2018 Order out of chaos: slowly reversing mean flows emerge from turbulently generated internal waves. Phys. Rev. Lett. 120, 244505.
Favier, B., Godeferd, F. S. & Cambon, C. 2010 On space and time correlations of isotropic and rotating turbulence. Phys. Fluids 22 (1), 015101.
Garaud, P. 2018 Double-diffusive convection at low Prandtl number. Annu. Rev. Fluid Mech. 50 (1), 275298.
Goldreich, P. & Kumar, P. 1990 Wave generation by turbulent convection. Astrophys. J. 363, 694704.
Grisouard, N. & Bühler, O. 2012 Forcing of oceanic mean flows by dissipating internal tides. J. Fluid Mech. 708, 250278.
Kunze, E. 2017 Internal-wave-driven mixing: global geography and budgets. J. Phys. Oceanogr. 47 (6), 13251345.
Lecoanet, D., Le Bars, M., Burns, K. J., Vasil, G. M., Brown, B. P., Quataert, E. & Oishi, J. S. 2015 Numerical simulations of internal wave generation by convection in water. Phys. Rev. E 91 (6), 063016.
Lecoanet, D. & Quataert, E. 2013 Internal gravity wave excitation by turbulent convection. Mon. Not. R. Astron. Soc. 430 (3), 23632376.
Liot, O., Seychelles, F., Zonta, F., Chibbaro, S., Coudarchet, T., Gasteuil, Y., Pinton, J.-F., Salort, J. & Chillà, F. 2016 Simultaneous temperature and velocity Lagrangian measurements in turbulent thermal convection. J. Fluid Mech. 794, 655675.
Munroe, J. R. & Sutherland, B. R. 2014 Internal wave energy radiated from a turbulent mixed layer. Phys. Fluids 26 (9), 096604.
Pinçon, C., Belkacem, K. & Goupil, M. J. 2016 Generation of internal gravity waves by penetrative convection. Astron. Astrophys. 588 (A122), 121.
Rogers, T. M., Lin, D. N. C. & Lau, H. H. B. 2012 Internal gravity waves modulate the apparent misalignment of exoplanets around hot stars. Astrophys. J. Lett. 758 (1), L6.
Rogers, T. M., Lin, D. N. C., McElwaine, J. N. & Lau, H. H. B. 2013 Internal gravity waves in massive stars: angular momentum transport. Astrophys. J. 772 (1), 21.
Sano, M., Wu, X. Z. & Libchaber, A. 1989 Turbulence in helium-gas free convection. Phys. Rev. A 40, 64216430.
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34 (1), 559593.
Taylor, J. R. & Sarkar, S. 2007 Internal gravity waves generated by a turbulent bottom Ekman layer. J. Fluid Mech. 590, 331354.
Tennekes, H. 1975 Eulerian and Lagrangian time microscales in isotropic turbulence. J. Fluid Mech. 67 (3), 561567.
Thorpe, S. A. 2018 Models of energy loss from internal waves breaking in the ocean. J. Fluid Mech. 836, 72116.
Zhou, Y. & Rubinstein, R. 1996 Sweeping and straining effects in sound generation by high Reynolds number isotropic turbulence. Phys. Fluids 8 (3), 647649.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO
Type Description Title
Supplementary materials

Couston et al supplementary material
Movie of (a) $w(y = 0)$, (b) $T_z-\bar{T}_z$ at $y = 0$ (overbar denotes x average), (c) $w(z = 0.7)$, (d) $w(z = 1:3)$ for simulation case $C_8^{400}$. Variables in the wave region $(z > 1)$ in (a), (b) have been multiplied by $10^4$, $10^3$, respectively.

 Video (304.0 MB)
304.0 MB

The energy flux spectrum of internal waves generated by turbulent convection

  • Louis-Alexandre Couston (a1), Daniel Lecoanet (a2), Benjamin Favier (a1) and Michael Le Bars (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.