Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T07:01:14.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Third-order structure functions in rotating and stratified turbulence: a comparison between numerical, analytical and observational results

Published online by Cambridge University Press:  19 August 2014

Enrico Deusebio ,
P. Augier  and
E. Lindborg
Enrico Deusebio*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Linné Flow Centre, Department of Mechanics, Royal Institute of Technology (KTH), SE-10044 Stockholm, Sweden
P. Augier
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Linné Flow Centre, Department of Mechanics, Royal Institute of Technology (KTH), SE-10044 Stockholm, Sweden
E. Lindborg
Affiliation:
Linné Flow Centre, Department of Mechanics, Royal Institute of Technology (KTH), SE-10044 Stockholm, Sweden
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

First, we review analytical and observational studies on third-order structure functions including velocity and buoyancy increments in rotating and stratified turbulence and discuss how these functions can be used in order to estimate the flux of energy through different scales in a turbulent cascade. In particular, we suggest that the negative third-order velocity–temperature–temperature structure function that was measured by Lindborg & Cho (Phys. Rev. Lett., vol. 85, 2000, p. 5663) using stratospheric aircraft data may be used in order to estimate the downscale flux of available potential energy (APE) through the mesoscales. Then, we calculate third-order structure functions from idealized simulations of forced stratified and rotating turbulence and compare with mesoscale results from the lower stratosphere. In the range of scales with a downscale energy cascade of kinetic energy (KE) and APE we find that the third-order structure functions display a negative linear dependence on separation distance $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}} r $, in agreement with observation and supporting the interpretation of the stratospheric data as evidence of a downscale energy cascade. The spectral flux of APE can be estimated from the relevant third-order structure function. However, while the sign of the spectral flux of KE is correctly predicted by using the longitudinal third-order structure functions, its magnitude is overestimated by a factor of two. We also evaluate the third-order velocity structure functions that are not parity invariant and therefore display a cyclonic–anticyclonic asymmetry. In agreement with the results from the stratosphere, we find that these functions have an approximate $ r^{2} $-dependence, with strong dominance of cyclonic motions.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Turbulent Flows: Turbulent Flows Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 755 , 25 September 2014 , pp. 294 - 313
Copyright
© 2014 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

Augier, P., Billant, P. & Chomaz, J. M. 2013 Stratified turbulence forced with columnar dipoles. Numerical study. J. Fluid Mech. (submitted).Google Scholar
Augier, P., Billant, P., Negretti, M. E. & Chomaz, J. M. 2014 Experimental study on stratified turbulence forced with columnar dipoles. Phys. Fluids 26, 046603.Google Scholar
Augier, P., Galtier, S. & Billant, P. 2012 Kolmogorov laws for stratified turbulence. J. Fluid Mech. 709, 659–670.CrossRefGoogle Scholar
Augier, P. & Lindborg, E. 2013 A new formulation of the spectral energy budget of the atmosphere, with application to two high-resolution general circulation models. J. Atmos. Sci. 70 (70), 2293–2308.Google Scholar
Babin, A., Mahalov, A., Nicolaenko, B. & Zhou, Y. 1997 On the asymptotic regimes and the strongly stratified limit of rotating Boussinesq equations. J. Theor. Comput. Fluid Dyn. 9 (3–4), 223–251.Google Scholar
Bartello, P. 1995 Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atmos. Sci. 52, 4410–4428.Google Scholar
Bartello, P., Metais, O. & Lesieur, M. 1994 Coherent structures in rotating three-dimensional turbulence. J. Fluid Mech. 273, 1–30.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13 (6), 1645–1651.Google Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343–368.Google Scholar
Burgess, B. H., Erler, A. R. & Shepherd, T. G. 2013 The troposphere-to-stratosphere transition in kinetic energy spectra and nonlinear spectral fluxes as seen in ECMWF analyses. J. Atmos. Sci. 70 (2), 669–687.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Charney, J. G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28 (6), 1087–1094.Google Scholar
Charney, J. G. & Drazin, P. G. 1961 Propagation of planetary-scale disturbances from the lower into the upper atmosphere. J. Geophys. Res. 66 (1), 83–109.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.CrossRefGoogle Scholar
Deusebio, E., Vallgren, A. & Lindborg, E. 2013 The route to dissipation in strongly stratified and rotating flows. J. Fluid Mech. 720, 66–103.CrossRefGoogle Scholar
Dewan, E. M. 1979 Stratospheric wave spectra resembling turbulence. Science 204 (4395), 832–835.Google Scholar
Embid, P. F. & Majda, A. J. 1998 Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers. J. Geophys. Astrophys. Fluid Dyn. 87 (1–2), 1–50.CrossRefGoogle Scholar
Hopfinger, E. J. & van Heijst, G. J. F. 1993 Vortices in rotating fluids. Annu. Rev. Fluid Mech. 25, 241–289.Google Scholar
Kolmogorov, A. N. 1941a Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 19–21 (in Russian).Google Scholar
Kolmogorov, A. N. 1941b The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303 (in Russian).Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10 (7), 1417–1423.Google Scholar
Laval, J. P., McWilliams, J. C. & Dubrulle, B. 2003 Forced stratified turbulence: Successive transitions with Reynolds number. Phys. Rev. E 68, 036308.Google Scholar
Lilly, D. K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40 (3), 749–761.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, L01809.Google Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.CrossRefGoogle Scholar
Lindborg, E. 2007 Horizontal wavenumber spectra of vertical vorticity and horizontal divergence in the upper troposphere and lower stratosphere. J. Atmos. Sci. 64, 1017–1025.Google Scholar
Lindborg, E. & Brethouwer, G. 2007 Stratified turbulence forced in rotational and divergent modes. J. Fluid Mech. 586, 83–108.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.CrossRefGoogle ScholarPubMed
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
Lorenz, E. N. 1955 Available potential energy and the maintenance of the general circulation. Tellus 7 (2), 157–167.Google Scholar
Moisy, F., Morize, C., Rabaud, M. & Sommeria, J. 2011 Decay laws, anisotropy and cyclone–anticyclone asymmetry in decaying rotating turbulence. J. Fluid Mech. 666, 5–35.Google Scholar
Nastrom, G. D. & Gage, K. S. 1985 A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci. 42, 950–960.Google Scholar
Riley, J. J. & deBruynKops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15 (7), 2047–2059.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
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
Vallgren, A., Deusebio, E. & Lindborg, E. 2011 A possible explanation of the atmospheric kinetic and potential energy spectra. Phys. Rev. Lett. 99, 99–101.Google Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation (Electronic Version). Cambridge University Press.Google Scholar
Waite, M. L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281–308.Google Scholar
Waite, M. L. & Bartello, P. 2006 The transition from geostrophic to stratified turbulence. J. Fluid Mech. 568, 89–108.Google Scholar
Waite, M. L. & Snyder, C. 2009 The mesoscale kinetic energy spectrum of a baroclinic life cycle. J. Atmos. Sci. 66, 883–901.Google Scholar
Waite, M. L. & Snyder, C. 2013 Mesoscale energy spectra of moist baroclinic waves. J. Atmos. Sci. 70 (4), 1242–1256.Google Scholar
Xia, H., Byrne, D., Falkovich, G. & Shats, M. 2011 Upscale energy transfer in thick turbulent fluid layers. Nat. Phys. 7 (4), 321–324.Google Scholar