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Structural stability theory of two-dimensional fluid flow under stochastic forcing

Published online by Cambridge University Press:  15 July 2011

NIKOLAOS A. BAKAS  and
PETROS J. IOANNOU
NIKOLAOS A. BAKAS*
Affiliation:
Department of Physics, National and Kapodistrian University of Athens, Office 32, Building IV, Panepistimiopolis, 15784 Athens, Greece
PETROS J. IOANNOU
Affiliation:
Department of Physics, National and Kapodistrian University of Athens, Office 32, Building IV, Panepistimiopolis, 15784 Athens, Greece
*
†Email address for correspondence: nikos.bakas@gmail.com
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Abstract

Large-scale mean flows often emerge in turbulent fluids. In this work, we formulate a stability theory, the stochastic structural stability theory (SSST), for the emergence of jets under external random excitation. We analytically investigate the structural stability of a two-dimensional homogeneous fluid enclosed in a channel and subjected to homogeneous random forcing. We show that two generic competing mechanisms control the instability that gives rise to the emergence of an infinitesimal jet: advection of the eddy vorticity by the mean flow that is shown to be jet forming and advection of the vorticity gradient of the jet by the eddies that is shown to hinder the formation of the mean flow. We show that stochastic forcing with small streamwise coherence and an amplitude larger than a certain threshold leads to the emergence of jets in the channel through a bifurcation of the non-linear SSST system.

JFM classification

Mathematical Foundations: General fluid mechanics Instability: Instability Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 682 , 10 September 2011 , pp. 332 - 361
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

‡

Present address: National and Kapodistrian University of Athens, Build. IV office 32, Panepistimiopolis, Zografos, Athens, Greece.

References

REFERENCES

Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13, 3258–3269.CrossRefGoogle Scholar
Berloff, P., Kamenkovich, I. & Pedlosky, J. 2009 a A mechanism of formation of multiple zonal jets in the oceans. J. Fluid Mech. 628, 395–425.CrossRefGoogle Scholar
Berloff, P., Kamenkovich, I. & Pedlosky, J. 2009 b A model of multiple zonal jets in the oceans: dynamical and kinematical analysis. J. Phys. Oceanogr. 39, 2711–2734.CrossRefGoogle Scholar
Bernstein, J. & Farrell, B. F. 2010 Low-frequency variability in a turbulent baroclinic jet: Eddy-mean flow interactions in a two-level model. J. Atmos. Sci. 67, 452–467.CrossRefGoogle Scholar
Bouchet, F. & Sommeria, J. 2002 Emergence of intense jets and Jupiter's Great Red Spot as maximum-entropy structures. J. Fluid Mech. 464, 165–207.CrossRefGoogle Scholar
Brewer, J. W. 1978 Kronecker products and matrix calculus in system theory. IEEE Trans. Circuits Syst. 9, 772–781.CrossRefGoogle Scholar
Charney, J. G. & DeVore, J. G. 1979 Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci. 36, 1205–1216.2.0.CO;2>CrossRefGoogle Scholar
Connaughton, C., Nadiga, B., Nazarenko, S. & Quinn, B. 2010 Modulational instability of Rossby and drift waves and generation of zonal jets. J. Fluid Mech. 654, 207–231.CrossRefGoogle Scholar
Davis, P. J. 1978 Circulant Matrices. Wiley-Interscience.Google Scholar
DelSole, T. 1996 Can quasigeostrophic turbulence be modeled stochastically? J. Atmos. Sci. 53, 1617–1633.2.0.CO;2>CrossRefGoogle Scholar
DelSole, T. 1999 Stochastic models of shear-flow turbulence with enstrophy transfer to subgrid scales. J. Atmos. Sci. 56, 3692–3703.2.0.CO;2>CrossRefGoogle Scholar
DelSole, T. 2001 A theory for the forcing and dissipation in stochastic turbulence models. J. Atmos. Sci. 58, 3762–3775.2.0.CO;2>CrossRefGoogle Scholar
DelSole, T. 2004 Stochastic models of quasigeostrophic turbulence. Surv. Geophys. 25, 107–194.CrossRefGoogle Scholar
DelSole, T. & Farrell, B. F. 1995 A stochastically excited linear system as a model for quasigeostrophic turbulence: Analytic results for one- and two-layer fluids. J. Atmos. Sci. 52, 2531–2547.2.0.CO;2>CrossRefGoogle Scholar
DelSole, T. & Farrell, B. F. 1996 The quasi-linear equilibration of a thermally maintained, stochastically excited jet in a quasigeostrophic model. J. Atmos. Sci. 53, 1781–1797.2.0.CO;2>CrossRefGoogle Scholar
Diamond, P. H., Itoh, S. I., Itoh, K. & Hahm, T. S. 2005 Zonal flows in plasma – a review. Plasma Phys. Control. Fusion 47, R35–R161.CrossRefGoogle Scholar
Dijkstra, H. A. & Katsman, C. A. 1997 Temporal variability of the wind driven quasi-geostrophic double gyre ocean circulation: basic bifurcation diagrams. Geophys. Astrophys. Fluid Dyn. 53, 195–232.CrossRefGoogle Scholar
Duguet, Y., Schlatter, P. & Henningson, D. S. 2009 Localized edge states in plane Couette flow. Phys. Fluids 21, 111701.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.CrossRefGoogle ScholarPubMed
Farrell, B. F. & Ioannou, P. J. 1993 a Stochastic dynamics of baroclinic waves. J. Atmos. Sci. 50, 4044–4057.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993 b Stochastic forcing of perturbation variance in unbounded shear and deformation flows. J. Atmos. Sci. 50, 200–211.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993 c Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluid. 5, 2600–2609.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1994 A theory for the statistical equilibrium energy spectrum and heat flux produced by transient baroclinic waves. J. Atmos. Sci. 51, 2685–2698.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1995 Stochastic dynamics of the midlatitude atmospheric jet. J. Atmos. Sci. 52, 1642–1656.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part I: Autonomous operators. J. Atmos. Sci. 53, 2025–2040.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1998 Perturbation structure and spectra in turbulent channel flow. Theor. Comput. Fluid Dyn. 11, 215–227.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2003 Structural stability of turbulent jets. J. Atmos. Sci. 60, 2101–2118.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64, 3652–3655.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2008 Formation of jets in baroclinic turbulence. J. Atmos. Sci. 65, 3352–3355.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2009 a Emergence of jets from turbulence in the shallow-water equations on an equatorial beta-plane. J. Atmos. Sci. 66, 3197–3207.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2009 b A stochastic structural stability theory model of the drift wave-zonal flow system. Phys. Plasma. 16, 112903.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2009 c A theory of baroclinic turbulence. J. Atmos. Sci. 66, 2444–2454.CrossRefGoogle Scholar
Fujisawa, A., Itoh, K., Shimizu, A., Nakano, H., Ohshima, S., Iguchi, H., Matsuoka, K., Okamura, S., Minami, T., Yoshimura, Y., Nagaoka, K., Ida, K., Toi, K., Takahashi, C., Kojima, M., Nishimura, S., Isobe, M., Suzuki, C., Akiyama, T., Ido, T., Nagashima, Y., Itoh, S.-I. & Diamond, P. H. 2008 Experimental studies of zonal flow and field in compact helical system plasma. Phys. Plasmas 15, 055906.CrossRefGoogle Scholar
Gill, A. E. 1974 The stability of planetary waves on an infinite beta plane. Geophys. Fluid Dyn. 6, 29–47.CrossRefGoogle Scholar
Graham, A. 1981 Kronecker Products and Matrix Calculus with Applications. Ellis Horwood Ltd.Google Scholar
Huang, H. P., Galperin, B. H. & Sukoriansky, S. 2001 Anisotropic spectra in two-dimensional turbulence on the surface of a rotating sphere. Phys. Fluids 13, 225–240.CrossRefGoogle Scholar
Huang, H. P. & Robinson, W. A. 1998 Two-dimensional turbulence and persistent zonal jets in a global barotropic model. J. Atmos. Sci. 55, 611–632.2.0.CO;2>CrossRefGoogle Scholar
Hunt, J. C. R. & Corruthers, D. J. 1990 Rapid distortion theory and the ‘problems of turbulence. J. Fluid Mech. 212, 497–532.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 Amplification of coherent structures in the turbulent Couette flow: an input–output analysis at low Reynolds number. J. Fluid Mech. 643, 333–348.CrossRefGoogle Scholar
Ingersoll, A. P. 1990 Atmospheric dynamics of the outer planets. Science 248, 308–315.CrossRefGoogle ScholarPubMed
Ingersoll, A. P., Gierasch, P. J., Banfield, D. & Vasavada, A. R. 2000 Moist convection as an energy source for the large-scale motions in Jupiter's atmosphere. Nature 403, 630–632.CrossRefGoogle ScholarPubMed
Jeffreys, H. 1926 On the dynamics of geostrophic winds. Q. J. R. Meteorol. Soc. 52, 85–104.CrossRefGoogle Scholar
Jovanovic, M. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.CrossRefGoogle Scholar
Kitamura, Y. & Ishioka, K. 2007 Equatorial jets in decaying shallow-water turbulence on a rotating sphere. J. Atmos. Sci. 64, 3340–3353.CrossRefGoogle Scholar
Krishnamurti, R. & Howard, L. N. 1981 Large-scale flow generation in turbulent convection. Proc. Natl Acad. Sci. 78, 1981–1985.CrossRefGoogle ScholarPubMed
Kuo, H.-L. 1951 Vorticity transfer as related to the development of the general circulation. J. Meteorol. 8, 307–315.2.0.CO;2>CrossRefGoogle Scholar
Laval, J.-P., Dubrulle, B. & McWilliams, J. C. 2003 Langevin models of turbulence: Renormalization group, distant interaction algorithms or rapid distortion theory? Phys. Fluids 15, 1327–1339.CrossRefGoogle Scholar
Legras, B. & Ghil, M. 1985 Persistent anomalies, blocking and variations in atmospheric predictability. J. Atmos. Sci. 42, 433–471.2.0.CO;2>CrossRefGoogle Scholar
Lorenz, E. N. 1974 Barotropic instability of Rossby wave motion. J. Atmos. Sci. 29, 258–264.2.0.CO;2>CrossRefGoogle Scholar
Marshall, J. & Molteni, F. 1993 Toward a dynamical understanding of planetary-scale flow regimes. J. Atmos. Sci. 50, 1792–1818.2.0.CO;2>CrossRefGoogle Scholar
Nazarenko, S. & Quinn, B. 2009 Triple cascade behavior in quasigeostrophic and drift turbulence and generation of zonal jets. Phys. Rev. Lett. 103, 118501.CrossRefGoogle ScholarPubMed
Newman, M., Sardeshmukh, P. D. & Penland, C. 1997 Stochastic forcing of the wintertime extratropical flow. J. Atmos. Sci. 54, 435–455.2.0.CO;2>CrossRefGoogle Scholar
Nozawa, T. & Yoden, Y. 1997 Formation of zonal band structure in forced two-dimensional turbulence on a rotating sphere. Phys. Fluids 9, 2081–2093.CrossRefGoogle Scholar
Pierrehumbert, R. & Malguzzi, P. 1984 Forced coherent structures and local multiple equilibria in a barotropic atmosphere. J. Atmos. Sci. 41, 246–257.2.0.CO;2>CrossRefGoogle Scholar
Rayleigh, Lord 1880 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 9, 57.Google Scholar
Read, P. L., Yamazaki, Y. H., Lewis, S. R., Williams, P. D., Miki-Yamazaki, K., Sommeria, J., Didelle, H. & Fincham, A. 2004 Jupiter's and Saturn's convectively driven banded jets in the laboratory. Geophys. Res. Lett. 87, 1961–1967.Google Scholar
Read, P. L., Yamazaki, Y. H., Lewis, S. R., Williams, P. D., Wordsworth, R. & Miki-Yamazaki, K. 2007 Dynamics of convectively driven banded jets in the laboratory. J. Atmos. Sci. 64, 4031–4052.CrossRefGoogle Scholar
Rhines, P. B. 1975 Waves and turbulence on a beta plane. J. Fluid Mech. 69, 417–433.CrossRefGoogle Scholar
Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310.CrossRefGoogle Scholar
Salyk, C., Ingersoll, A. P., Lorre, J., Vasavada, A. & Del Genio, A. D. 2006 Interaction between eddies and mean flow in Jupiter's atmosphere: Analysis of Cassini imaging data. Icarus 185, 430–442.CrossRefGoogle Scholar
Shepherd, T. G. 1987 A spectral view of nonlinear fluxes and stationary–transient interaction in the atmosphere. J. Atmos. Sci. 44, 1166–1178.2.0.CO;2>CrossRefGoogle Scholar
Simonnet, E., Ghil, M. & Dijkstra, H. A. 2005 Homoclinic bifurcations in the barotropic quasi-geostrophic double-gyre circulation. J. Mar. Res. 63, 931–956.CrossRefGoogle Scholar
Starr, V. 1968 Physics of Negative Viscosity Phenomena. McGraw Hill.Google Scholar
Vallis, G. K. & Maltrud, M. E. 1993 Generation of mean flows and jets on a beta plane and over topography. J. Phys. Oceanogr. 23, 1346–1362.2.0.CO;2>CrossRefGoogle Scholar
Vasavada, A. R. & Showman, A. P. 2005 Jovian atmospheric dynamics. An update after Galileo and Cassini. Rep. Prog. Phys. 68, 1935–1996.CrossRefGoogle Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.CrossRefGoogle Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
Whitaker, J. S. & Sardeshmukh, P. D. 1998 A linear theory of extratropical synoptic eddy statistics. J. Atmos. Sci. 55, 237–258.2.0.CO;2>CrossRefGoogle Scholar
Zhang, Y. & Held, I. M. 1999 A linear stochastic model of a GCM's midlatitude storm tracks. J. Meteorol. Soc. Japan 56, 3416–3435.Google Scholar