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Wave-vortex dynamics in rotating shallow water

Published online by Cambridge University Press:  26 April 2006

Marie Farge  and
Robert Sadourny
Marie Farge
Affiliation:
Laboratoire de Météorologie Dynamique du CNRS, 24, rue Lhomond, 75231 Paris Cedex 05, France
Robert Sadourny
Affiliation:
Laboratoire de Météorologie Dynamique du CNRS, 24, rue Lhomond, 75231 Paris Cedex 05, France
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Abstract

We investigate how two-dimensional turbulence is modified when the incompressibility constraint is removed, by numerically integrating the full Saint-Venant (shallow-water) equations. In the case of small geopotential fluctuations considered here, we find no energy exchange between the inertio-gravitational and the potentio-vortical components of the flow. At small scales, the potentio-vortical component behaves as if the flow were incompressible, while we observe an intense direct energy cascade within the inertio-gravitational component. At large scales, the reverse potentio-vortical energy cascade is reduced when the level of inertio-gravitational energy is high. Looking at the effect of rotation, we find that a fast rotation rate tends to inhibit all three cascades. In particular, the inhibition of the inertio-gravitational energy cascade towards small scales implies that the geostrophic adjustment process is hindered by an increase of rotation. Concerning the structure of the coherent vortices emerging out of these decaying turbulent flows, we observe that the smallest scales are concentrated inside the vortex cores and not on their periphery.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 206 , September 1989 , pp. 433 - 462
Copyright
© 1989 Cambridge University Press

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References

Asselin, R. 1972 Frequency filter for time integrations. Mon. Weather Rev. 100, 487–490.Google Scholar
Babiano, A., Basdevant, C., Legras, B. & Sadourny, R. 1984 Dynamiques comparées du tourbillon et d'un scalaire passif en turbulence bidimensionnelle incompressible. C.R. Acad. Sci. Paris 299 II, 601–604.Google Scholar
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
Bardina, J., Ferziger, J. H. & Rogallo, R. S. 1985 Effect of rotation on isotropic turbulence: computation and modelling. J. Fluid Mech. 154, 321–336.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
Basdevant, C. & Sadourny, R. 1975 Ergodic properties of inviscid truncated models of two-dimensional incompressible flows. J. Fluid Mech. 69, 673–688.Google Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids Suppl. II, 12, 233–239.Google Scholar
Bennett, A. F. & Haidvogel, D. B. 1983 Low-resolution numerical simulation of decaying two-dimensional turbulence. J. Atmos. Sci. 40, 738–748.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
Cahn, A. 1945 An investigation of the free oscillation of a simple current system. J. Met. 2, 113–119.Google Scholar
Charney, J. 1971 Geostrophic turbulence. J. Atmos. Sci. 28, 1087–1095.Google Scholar
Couder, Y. 1984 Two-dimensional grid turbulence in a thin liquid film. J. Phys. Lett. 45, 353–360.Google Scholar
Craya, A. 1958 Contribution à l'analyse de la turbulence associée à des vitesses moyennes. P.S.T. Ministère de l'Air, p. 345.Google Scholar
Desbois, M. 1975 Large-scale kinetic energy spectra from Eulerian analysis of Eole wind data. J. Atmos. Sci. 32, 1838–1847.Google Scholar
Dritschel, D. G. 1986 The nonlinear evolution of rotating configurations of uniform vorticity. J. Fluid Mech. 172, 157–182.Google Scholar
Errico, R. M. 1981 An analysis of interactions between geostrophic and ageostrophic modes in a simple model. J. Atmos. Sci. 38, 544–553.Google Scholar
Errico, R. M. 1984 The statistical equilibria solution of a primitive equations model. Tellus 36A, 42–51.Google Scholar
Farge, M. 1987 Normalization of high-resolution raster display applied to turbulent fields. In Advances in Turbulence (ed. G. Comte-Bellot), pp. 111–123. Springer.
Farge, M. 1988 Nonlinear dynamics of inertio-gravity waves. In 59th Varenna Course on Nonlinear Topics in Ocean Physics (ed. A. Osborne). North-Holland.
Farge, M. & Lacarra, J. F. 1988 The numerical modelling of shallow water equations. J. Méc. Théor. Appl., Special Issue, Suppl. 2 to 7, 63–86.Google Scholar
Farge, M. & Rabreau, G. 1988 Wavelet transform to detect and analyse coherent structures in two-dimensional turbulent flows. C. R. Acad. Sci. Paris 307II, 1479–1486.Google Scholar
Farge, M. & Sadourny, R. 1986a Inhibitation de la turbulence bi-dimensionnelle par une rotation d'entrénement. C.R. Acad. Sci. Paris 302II, 847–850.Google Scholar
Farge, M. & Sadourny, R. 1986b Effets des ondes d'inertie-gravité sur une turbulence bi-dimensionnelle non forcée en rotation. C.R. Acad. Sci. Paris 303II, 881–886.Google Scholar
Feiereisen, W. J., Reynolds, W. C. & Ferziger, J. H. 1981 Numerical simulation of a compressible homogeneous turbulent shear flow. Mechanical Engineering Department, Rep. TF-13. Stanford University.
Fornberg, B. 1977 A numerical study of two-dimensional turbulence. J. Comput. Phys. 25, 1–31.Google Scholar
Fox, D. G. & Orszag, S. H. 1973 Inviscid dynamics of two-dimensional turbulence. Phys. Fluids 16, 167–171.Google Scholar
Herring, J. R. 1974 Approach of axisymmetric turbulence to isotropy. Phys. Fluids 17, 859–872.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
Herring, J. R., McWilliams, J. C., Métais, O. & Gamage, N. 1987 Vortical turbulence in a stratified fluid. Third Int. Symp. on Stratified Flows, IAHR meeting, Pasadena, February 3-5. Preprints.
Holloway, G. 1983 Effects of planetary wave propagation and finite depth on the predictability of atmospheres. J. Atmos. Sci. 40, 314–327.Google Scholar
Hopfinger, E. J. 1983 The structure of turbulence in homogeneous and stratified rotating fluids. J. Méc. Special Issue on Two-Dimensional Turbulence, pp. 21–44.Google Scholar
Hoyer, J. M. & Sadourny, R. 1982 Inhibition of baroclinic instability of low-resolution models. J. Atmos. Sci. 39, 2138–2143.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges of two-dimensional turbulence. Phys. Fluids 10, 1417–1423.Google Scholar
Leith, C. E. 1968 Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11, 617–673.Google Scholar
Leith, C. E. 1980 Nonlinear normal mode initialization and quasi-geostrophic theory. J. Atmos. Sci. 37, 958–968.Google Scholar
Lilly, D. K. 1983 Stratified turbulence and mesoscale variability of the atmosphere. J. Atmos. Sci. 40, 749–761.Google Scholar
Lorenz, E. N. 1980 Attractor sets and quasi-geostrophic equilibrium. J. Atmos. Sci. 37, 1685–1699.Google Scholar
Lorenz, E. N. 1986 On the existence of a slow manifold. J. Atmos. Sci. 43, 1547–1557.Google Scholar
McWilliams, J. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.Google Scholar
Métais, O. & Herring, J. R. 1989 Numerical experiments in forced stably stratified turbulence. J. Fluid Mech. 202, 97–115.Google Scholar
Morel, P. & Larchevéque, M. 1974 Relative dispersion of constant-level balloons in the 200-mb general circulation. J. Atmos. Sci. 31, 2189–2196.Google Scholar
Moyal, J. E. 1952 The spectra of turbulence in a compressible fluid eddy turbulence and random noise. Proc. Camb. Phil. Soc. 48, 329.Google Scholar
Müller, P., Lien, R. C. & Williams, R. 1988 Estimate of potential vorticity at small scales in the ocean. J. Phys. Oceanogr. 18, 401–416.Google Scholar
Obukhov, A. 1949 On the question of geostrophic winds. Izv. Akad. Nauk. SSSR, Geogra. Geofiz. 13, 281–306.Google Scholar
Riabouchinsky, D. 1932 Sur l'analogie hydraulique des mouvements d'un fluide compressible. C.R. Acad. Sci. Paris 195, 998.Google Scholar
Rhines, P. B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–443.Google Scholar
Rhines, P. B. 1979 Geostrophie turbulence. Ann. Rev. Fluid Mech. 11, 401–441.Google Scholar
Riley, J. J., Metcalfe, R. W. & Weissman, M. A. 1981 Direct numerical simulations of homogeneous turbulence in density-stratified fluids. Proc. AIP Conf. on Nonlinear Properties of Internal Waves, pp. 79–112.
Roy, P. 1986 Simulation numérique d'un champ turbulent homogène incompressible soumis à des gradients, de vitesse moyenne. Thè d'état, Université de Nice.
Sadourny, R. 1975 The dynamics of finite-difference models of the shallow-water equations. J. Atmos. Sci. 32, 680–689.Google Scholar
Sadourny, R. 1985 Quasigeostrophie turbulence. In 88th Varenna Course on Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (ed. M. Ghil). North-Holland.
Sadourny, R. & Basdevant, C. 1985 Parametrization of sub-grid scale barotropic and baroclinie eddies in quasi-geostrophie models: the antieipated potential vorticity method. J. Atmos. Sci. 42, 1353–1363.Google Scholar
Saffman, P. G. 1971 On the spectrum and decay of random two-dimensional vorticity distributions at large Reynolds number. Stud. Appl. Maths 50, 377–383.Google Scholar
Salmon, R. 1978 Two-layer quasi-geostrophic turbulence in a simple special case. Geophys. Astrophys. Fluid Dyn. 10, 25–51.Google Scholar
Sommeria, J. 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139–168.Google Scholar
Temperton, C. 1983 Very fast real Fourier transforms. In Special Topics and Applied Mathematics, pp. 165–171. North-Holland.
Warn, T. 1986 Statistical mechanical equilibria of the shallow-water equations. Tellus 38A, 1–11.Google Scholar
Wigeland, R. A. & Nagib, H. M. 1978 Grid-generated turbulence with and without rotation about the streamwise direction. 11th Fluid and Heat Transfer Rep. R. 78-1. Illinois Institute of Technology.
Zabusky, N. J. 1984 Contour dynamics: a method for inviseid and nearly inviseid two-dimensional flows. Proc. IUTAM Symp. on Turbulence and Chaotic Phenomena (ed. T. Tatsumi), pp. 251–257. North-Holland.
Zakharov, V. E. & Sagdeev, R. Z. 1970 Spectrum of acoustic turbulence. Sov. Phys. Dokl. 15, 439–441.Google Scholar