Hostname: page-component-7bb8b95d7b-wpx69 Total loading time: 0 Render date: 2024-09-20T05:11:58.820Z Has data issue: false hasContentIssue false
  • English
  • Français

On the generation of large-scale structures in a homogeneous eddy field

Published online by Cambridge University Press:  16 November 2010

TIMOUR RADKO
TIMOUR RADKO*
Affiliation:
Department of Oceanography, Naval Postgraduate School, Monterey, CA 9394, USA
*
†Email address for correspondence: tradko@nps.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

An analytical theory is developed which illustrates dynamics of the spontaneous generation of large-scale structures in the unforced two-dimensional eddying flows. The eddy field is represented by the closely packed array of standing coherent vortices whose intensity is weakly modulated by the long-wavelength perturbations introduced into the system. The asymptotic multiscale analysis makes it possible to identify instabilities resulting from the positive feedback of the background eddies on large-scale perturbations. Initially, these instabilities amplify at a rate proportional to the square root of their wavenumber. Linear growth is arrested when the amplitude of the long-wavelength perturbations reaches the level of background eddies. The subsequent evolutionary pattern is characterized by the emergence of relatively sharp features in the large-scale streamfunction field – features suggestive of the coherent jets commonly observed in eddying geophysical flows. The proposed solutions differ substantially from their counterparts in forced-dissipative systems, exemplified by the canonical model of Kolmogorov flow. The asymptotic model is successfully tested against numerical simulations.

JFM classification

Instability: Instability Geophysical and Geological Flows: Ocean processes Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 668 , 10 February 2011 , pp. 76 - 99
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Baldwin, M., Rhines, P., Huang, H.-P. & McIntyre, M. 2007 The jet-stream conundrum. Science 315, 467–468.CrossRefGoogle ScholarPubMed
Balmforth, N. J. & Young, Y.-N. 2002 Stratified Kolmogorov flow. J. Fluid Mech. 450, 131–167.CrossRefGoogle Scholar
Balmforth, N. J. & Young, Y.-N. 2005 Stratified Kolmogorov flow. Part 2. J. Fluid Mech. 528, 23–42.CrossRefGoogle Scholar
Berloff, P., Kamenkovich, I. & Pedlosky, J. 2009 a A mechanism for the 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
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1987 Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics, Springer.Google Scholar
Drazin, P. G. 1977 On the instability of an internal gravity wave. Proc. R. Soc. Lond. A 356, 411–432.Google Scholar
Drazin, P. G. 1992 Nonlinear Systems. Cambridge Texts in Applied Mathematics, Cambridge University Press.CrossRefGoogle Scholar
Dritschel, D. & McIntyre, M. 2008 Multiple jets as PV staircases: the Phillips effect and the resilience of eddy-transport barriers. J. Atmos. Sci. 65, 855–874.CrossRefGoogle Scholar
Frisch, U., Legras, B. & Villone, B. 1996 Large scale Kolmogorov flow on the beta-plane and resonant wave interaction. Physica D 94, 36–56.CrossRefGoogle Scholar
Galperin, B., Nakano, H., Huang, H. & Sukoriansky, S. 2004 The ubiquitous zonal jets in the atmospheres of giant planets and Earth's oceans. Geophys. Res. Lett. 31, L13303.CrossRefGoogle Scholar
Gama, S., Vergassola, M. & Frisch, U. 1994 Negative eddy viscosity in isotropically forced two-dimensional flow: linear and nonlinear dynamics. J. Fluid Mech. 260, 95–126.CrossRefGoogle Scholar
Gill, A. E. 1974 The stability of planetary waves on an infinite beta-plane. Geophys. Fluid Dyn. 6, 29–47.CrossRefGoogle Scholar
Hogg, N. & Owens, B. 1999 Direct measurement of the deep circulation within the Brazil basin. Deep-Sea Res. 46, 335–353.Google Scholar
Huang, H.-P. & Robinson, W. 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
Kamenkovich, I., Berloff, P. & Pedlosky, J. 2009 a Role of eddy forcing in the dynamics of zonal jets in the north Atlantic. J. Phys. Oceanogr. 39, 1361–1379.CrossRefGoogle Scholar
Kamenkovich, I., Berloff, P. & Pedlosky, J. 2009 b Anisotropic material transport by eddies and eddy-driven currents in a model of the North Atlantic. J. Phys. Oceanogr. 39, 3162–3175.CrossRefGoogle Scholar
Kaspi, I. & Flierl, G. 2007 Formation of jets by baroclinic instability on gas planet atmospheres. J. Atmos. Sci. 64, 3177–3194.CrossRefGoogle Scholar
Kevorkian, J. & Cole, J. D. 1996 Multiple Scale and Singular Perturbation Methods. Springer.CrossRefGoogle Scholar
Kraichnan, R. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.CrossRefGoogle Scholar
Kraichnan, R. & Montgomery, D. 1980 Two-dimensional turbulence. Rep. Prog. Phys. 43, 547–619.CrossRefGoogle Scholar
Legras, B. & Villone, B. 2009 Large-scale instability of a generalized turbulent Kolmogorov flow. Nonlinear Process. Geophys. 16, 569–577.CrossRefGoogle Scholar
Lorentz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.2.0.CO;2>CrossRefGoogle Scholar
Lorenz, E. N. 1972 Barotropic instability of Rossby wave motion. J. Atmos. Sci. 29, 258–269.2.0.CO;2>CrossRefGoogle Scholar
Manfroi, A. & Young, W. 1999 Slow evolution of zonal jets on the beta plane. J. Atmos. Sci. 56, 784–800.2.0.CO;2>CrossRefGoogle Scholar
Manfroi, A. & Young, W. 2002 Stability of beta-plane Kolmogorov flow. Physica D 162, 208–232.CrossRefGoogle Scholar
Maximenko, N., Bang, B. & Sasaki, H. 2005 Observational evidence of alternating zonal jets in the world ocean. Geophys. Res. Lett. 32, L12607.CrossRefGoogle Scholar
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.CrossRefGoogle Scholar
Meshalkin, L. & Sinai, Y. 1961 Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid. J. Appl. Math. Mech. 25, 1700–1705.CrossRefGoogle Scholar
Novikov, A. 2003 Modulational stability of cellular flows. Nonlinearity 16, 1607–1639.CrossRefGoogle Scholar
Novikov, A. & Papanicolau, G. 2001 Eddy viscosity of cellular flows. J. Fluid Mech. 446, 173–198.CrossRefGoogle Scholar
Panetta, L. 1993 Zonal jets in wide baroclinically unstable regions: persistence and scale selection. J. Atmos. Sci. 50, 2073–2106.2.0.CO;2>CrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.CrossRefGoogle Scholar
Radko, T. 2005 What determines the thickness of layers in a thermohaline staircase? J. Fluid Mech. 523, 79–98.CrossRefGoogle Scholar
Read, P. L., Yamazaki, Y. H., Lewis, S. R., Williams, P. D., Wordsworth, R., Miki-Yamazaki, K., Sommeria, J. & Didelle, H. 2007 Dynamics of convectively driven banded jets in the laboratory. J. Atmos. Sci. 64, 4031–4052.CrossRefGoogle Scholar
Rhines, P. B. 1994 Jets. Chaos 4, 313–339.CrossRefGoogle ScholarPubMed
Richards, K., Maximenko, N., Bryan, F. & Sasaki, H. 2006 Zonal jets in the Pacific ocean. Geophys. Res. Lett. 33, L03605.CrossRefGoogle Scholar
Sivashinsky, G. 1985 Weak turbulence in periodic flows. Physica D 17, 243–255.CrossRefGoogle Scholar
Shepherd, T. 1988 Nonlinear saturation of baroclinic instability. Part I. The two-layer model. J. Atmos. Sci. 45, 2014–2025.2.0.CO;2>CrossRefGoogle Scholar
Starr, V. P. 1968 Physics of Negative Viscosity Phenomena. McGraw-Hill.Google Scholar
Stern, M. E. & Radko, T. 1997 Maintaining the inshore shear of continental boundary currents. Dyn. Atmos. Oceans 27, 661–678.CrossRefGoogle Scholar
Treguier, A. & Panetta, L. 1994 Multiple zonal jets in a quasi-geostrophic model of the Antarctic circumpolar current. J. Phys. Oceanogr. 24, 2263–2277.2.0.CO;2>CrossRefGoogle Scholar
Vallis, G. & Maltrud, M. 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
Whitehead, J. A. 1975 Mean flow driven by circulation on a beta-plane. Tellus 27, 634–649.Google Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.CrossRefGoogle Scholar