Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-24T11:23:39.468Z Has data issue: false hasContentIssue false
  • English
  • Français

Energy spectra and coherent structures in forced two-dimensional and beta-plane turbulence

Published online by Cambridge University Press:  26 April 2006

M. E. Maltrud  and
G. K. Vallis
M. E. Maltrud
Affiliation:
Scripps Institution of Oceanography, La Jolla, CA 92093, USA
G. K. Vallis
Affiliation:
Institute of Marine Science and Institute of Nonlinear Science, University of California, Santa Cruz, CA 95064, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Results from a wide range of direct numerical simulations of forced-dissipative, differentially rotating two-dimensional turbulence are presented, in order to delineate the broad dependence of flow type on forcing parameters. For most parameter values the energy spectra of simulations forced at low wavenumbers are markedly steeper than the classical k−3 enstrophy inertial-range prediction, and although k−3 spectra can be produced under certain circumstances, the regime is not robust, and the Kolmogorov constant is not universal unless a slight generalization is made in the phenomenology. Long-lived, coherent vortices form in many cases, accompanied by steep energy spectra and a higher than Gaussian vorticity kurtosis. With the addition of differential rotation (the β-effect), a small number of fairly distinct flow regimes are observed. Coherent vortices weaken and finally disappear as the strength of the β-effect increases, concurrent with increased anisotropy and decreased kurtosis. Even in the absence of coherent vortices and with a Gaussian value of the kurtosis, the spectra remain relatively steep, although not usually as steep as for the non-rotating cases. If anisotropy is introduced at low wavenumbers, the anisotropy is transferred to all wavenumbers in the inertial range, where the dynamics are isotropic.

For those simulations that are forced at relatively high wavenumbers, a well resolved and very robust k−5/3 energy inertial range is observed, and the Kolmogorov constant appears universal. The low-wavenumber extent of the reverse energy cascade is essentially limited by the β-effect, which produces an effective barrier in wavenumber space at which energy accumulates, and by frictional effects which must be introduced to achieve equilibrium. Anisotropy introduced at large scales remains largely confined to the low wavenumbers, rather than being cascaded to small scales. When there is forcing at both large and small scales (which is of relevance to the Earth's atmosphere), energy and enstrophy inertial ranges coexist, with an upscale energy transfer and downscale enstrophy transfer in the same wavenumber interval, without the need for any dissipation mechanism between forcing scales.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 228 , July 1991 , pp. 321 - 342
Copyright
© 1991 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

Babiano, A., Basdevant, C., Legras, B. & Sadourny, R. 1987 Vorticity and passive-scalar dynamics in two-dimensional turbulence. J. Fluid Mech. 183, 379–397.Google Scholar
Basdevant, C., Legras, B., Sadourny, R. & Beland, M. 1981 A study of barotropic model flows: intermittency, waves and predictability. J. Atmos. Sci. 38, 2305–2326.Google Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in two-dimensional turbulence. Phys. Fluids Suppl. 12, II 233–239.Google Scholar
Benzi, R., Paladin, G., Patarnello, P., Santangelo, P. & Vulpiani, A. 1986 Intermittency and coherent structures in two-dimensional turbulence. J. Phys. A: Math. Gen. 19, 3771–3784.Google Scholar
Boer, G. J. & Shepherd, T. G. 1983 Large scale two-dimensional turbulence in the atmosphere. J. Atmos. Sci. 40, 164–184.Google Scholar
Fornberg, B. 1977 A numerical study of 2D turbulence. J. Comput. Phys. 25, 1–31.Google Scholar
Frisch, U. & Sulem, P.-L. 1984 Numerical simulation of the inverse cascade in two-dimensional turbulence. Phys. Fluids 27, 1921–1923.Google Scholar
Grant, H. L., Stewart, R. W. & Moillet, A. 1962 Turbulence spectra from a tidal channel. J. Fluid Mech. 12, 241–268.Google Scholar
Herring, J. R. 1975 Theory of two-dimensional anisotropic turbulence. J. Atmos. Sci. 32, 2254–2271.Google Scholar
Herring, J. R. & McWilliams, J. C. 1985 Comparison of direct numerical simulation of two-dimensional turbulence with two-point closure: the effects of intermittency. J. Fluid Mech. 153, 229–242.Google Scholar
Holloway, G. 1984 Contrary roles of planetary wave propagation in atmospheric predictability. In Proc. La Jolla Inst. Workshop on Predictability of Fluid Flows (ed. G. Holloway & B. West), pp. 593–600.
Holloway, G. & Hendershott, M. 1977 Stochastic closure for nonlinear Rossby waves. J. Fluid Mech. 82, 747–765.Google Scholar
Holloway, G. & Kristmannsson, S. 1984 Stirring and transport of tracer fields by geostrophic turbulence. J. Fluid Mech. 141, 27–50.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible fluid at very high Reynolds number. Dokl. Acad. Sci. USSR. 30, 299–303.Google Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.Google Scholar
Kraichnan, R. H. 1971 Inertial-range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47, 525–535.Google Scholar
Larsen, M. F., Kelley, M. C. & Gage, K. S. 1982 Turbulence spectra in the upper troposphere and lower stratosphere at periods between 2 hours and 40 days. J. Atmos. Sci. 39, 1035–1041.Google Scholar
Legras, B., Santangelo, P. & Benzi, R. 1988 High resolution numerical experiments for forced two-dimensional turbulence. Europhys. Lett. 5, 37–42.Google Scholar
Leith, C. E. 1968 Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11, 671–673.Google Scholar
Leith, C. E. & Kraichnan, R. H. 1972 Predictability of turbulent flows. J. Atmos. Sci. 29, 1041.Google Scholar
Lesieur, M. & Herring, J. 1985 Diffusion of a passive scalar in two-dimensional turbulence. J. Fluid. Mech. 161, 77–95.Google Scholar
Lilly, D. K. 1972 Numerical simulation studies of two-dimensional turbulence: Part 1. Models of statistically steady turbulence. Geophys. Fluid Dyn. 3, 290–319.Google Scholar
Lilly, D. K. 1989 Two-dimensional turbulence generated by energy sources at two scales. J. Atmos. Sci. 46, 2026–2030.Google Scholar
Mcwilliams, J. C. 1984 The emergence of coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.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
Patterson, G. S. & Orszag, S. A. 1971 Spectral calculation of isotropic turbulence: efficient removal of aliasing interaction. Phys. Fluids 14, 2538–2541.Google Scholar
Rhines, P. B. 1975 Waves and turbulence on the β-plane. J. Fluid Mech. 69, 417–443.Google Scholar
Sadourny, R. & Basdevant, C. 1985 Parameterization of subgrid-scale barotropic and baroclinic eddies in quasi-geostrophic models: Anticipated potential vorticity method. J. Atmos. Sci. 42, 1353–1363.Google Scholar
Salmon, R. 1982 Geostrophic turbulence In Topics in Ocean Physics, Proc. Intl School Phys. ‘Enrico Fermi’, Varenna, Italy, pp. 30–78.
Santangelo, P., Benzi, R. & Legras, B. 1989 The generation of vortices in high resolution, two-dimensional decaying turbulence and the influence of initial conditions on the breaking of self-similarity.. Phys. Fluids A 1, 1027–1034.Google Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. Mon. Weath. Rev. 91 (3), 99–164.Google Scholar
Vallis, G. K. 1985 Remarks on the predictability of two- and three-dimensional flow. Q. J. Met. Soc. 111, 1039–1047.Google Scholar
Vallis, G. K. & Hua, B. 1988 Eddy viscosity of the anticipated potential vorticity method. J. Atmos. Sci. 45, 617–627.Google Scholar
Yakhot, V. & Orszag, S. 1986 Renormalization group analysis of turbulence. 1. Basic theory. J. Sci. Comput. 1, 3–51.Google Scholar