Slow Evolution of Zonal Jets on the Beta Plane

A. J. Manfroi Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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W. R. Young Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Abstract

The authors study the stability of a barotropic sinusoidal meridional flow on a β plane. Because of bottom drag and lateral viscosity, the system is dissipative and forcing maintains a basic-state velocity that carries fluid across the planetary vorticity contours; this is a simple model of forced potential vorticity mixing. When the Reynolds number is slightly above the stability threshold, a perturbation expansion can be used to obtain an amplitude equation for the most unstable disturbances. These instabilities are zonal flows with a much larger length scale than that of the basic state.

Numerical and analytic considerations show that random initial perturbations rapidly reorganize into a set of fast and narrow eastward jets separated by slower and broader regions of westward flow. There then follows a much slower adjustment of the jets, involving gradual meridional migration and merger. Because of the existence of a Lyapunov functional for the dynamics, this one-dimensional inverse cascade ultimately settles into a steady solution.

For a fixed β, the meridional separation of the eastward jets depends on the bottom drag. When the bottom drag is zero, the process of jet merger proceeds very slowly to completion until only one jet is left in the domain. For small bottom drag, the steady-state meridional separation of the jets varies as (bottom drag)−1/3. Varying the nondimensional β parameter can change the instability from supercritical (when β is small) to subcritical (when β is larger). Thus, the system has a rich phenomenology involving multiple stable solutions, hysteritic transitions, and so on.

Corresponding author address: A. J. Manfroi, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093-0230.

Email: amanfroi@ucsd.edu

Abstract

The authors study the stability of a barotropic sinusoidal meridional flow on a β plane. Because of bottom drag and lateral viscosity, the system is dissipative and forcing maintains a basic-state velocity that carries fluid across the planetary vorticity contours; this is a simple model of forced potential vorticity mixing. When the Reynolds number is slightly above the stability threshold, a perturbation expansion can be used to obtain an amplitude equation for the most unstable disturbances. These instabilities are zonal flows with a much larger length scale than that of the basic state.

Numerical and analytic considerations show that random initial perturbations rapidly reorganize into a set of fast and narrow eastward jets separated by slower and broader regions of westward flow. There then follows a much slower adjustment of the jets, involving gradual meridional migration and merger. Because of the existence of a Lyapunov functional for the dynamics, this one-dimensional inverse cascade ultimately settles into a steady solution.

For a fixed β, the meridional separation of the eastward jets depends on the bottom drag. When the bottom drag is zero, the process of jet merger proceeds very slowly to completion until only one jet is left in the domain. For small bottom drag, the steady-state meridional separation of the jets varies as (bottom drag)−1/3. Varying the nondimensional β parameter can change the instability from supercritical (when β is small) to subcritical (when β is larger). Thus, the system has a rich phenomenology involving multiple stable solutions, hysteritic transitions, and so on.

Corresponding author address: A. J. Manfroi, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093-0230.

Email: amanfroi@ucsd.edu

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  • Balmforth, N. J., 1995: Solitary waves and homoclinic orbits. Annu. Rev. Fluid Mech.,27, 335–373.

  • Chapman, C. J., and M. R. E. Proctor, 1980: Nonlinear Rayleigh-Bénard convection with poorly conducting boundaries. J. Fluid Mech.,101, 759–782.

  • Fraerman, A. A., A. S. Mel’nikov, I. M. Nefedov, I. A. Shereshevskii, and A. V. Shpiro, 1997: Nonlinear relaxation dynamics in decomposing alloys: One-dimensional Cahn-Hilliard model. Phys. Rev. B,55, 6316–6323.

  • Frisch, U., B. Legras, and B. Villone, 1996: Large-scale Kolmogorov flow on the beta-plane and resonant wave interactions. Physica D,94, 36–56.

  • Gill, A. E., 1974: The stability of planetary waves on an infinite beta-plane. Geophys. Fluid Dyn.,6, 29–47.

  • Huang, H. P., and A. Robinson, 1998: Two-dimensional turbulence and persistent zonal jets in a global barotropic model. J. Atmos. Sci.,55, 611–632.

  • Kawasaki, K., and T. Ohta, 1982: Kink dynamics in one-dimensional nonlinear systems. Physica,116A, 573–593.

  • Lorenz, E. N., 1972. Barotropic instability of Rossby wave motion. J. Atmos. Sci.,29, 258–269.

  • Meshalkin, L. D., and I. G. Sinai, 1961: Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid. Appl. Math. Mech.,24, 1700–1705.

  • Nepomnyashchy, A. A., 1976: On the stability of the secondary flow of a viscous fluid in an infinite domain. Appl. Math. Mech.,40, 836–841.

  • Nozawa, T., and S. Yoden, 1997: Formation of zonal band structures in forced two-dimensional turbulence on a rotating sphere. Phys. Fluids,9, 2081–2093.

  • Panetta, R. L., 1993: Zonal jets in wide baroclinically unstable regions: Persistence and scale selection. J. Atmos. Sci.,50, 2073–2106.

  • Rhines, P. B., 1975: Waves and turbulence on a β-plane. J. Fluid Mech.,69, 417–443.

  • ——, 1994: Jets. Chaos,4, 313–339.

  • Salmon, R., 1980: Baroclinic instability and geostrophic turbulence. Geophys. Astrophys. Fluid Dyn.,15, 167–211.

  • Sivashinsky, G. I., 1985: Weak turbulence in periodic flows. Physica D,17, 243–255.

  • Thual, O., and S. Fauve, 1988: Localized structures generated by subcritical instabilities. J. Phys.,49, 1829–1833.

  • Vallis, G. K., and M. Maltrud, 1993. Generation of mean flow and jets on a beta-plane and over topography. J. Phys. Oceanogr.,23, 1346–1362.

  • Williams, G. P., 1978: Planetary circulations: I. Barotropic representation of Jovian and terrestrial turbulence. J. Atmos. Sci.,35, 1399–1426.

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