Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 17 1 2010 10.5194/npg-17-49-2010 http://www.nonlin-processes-geophys.net/17/49/2010/ http://www.nonlin-processes-geophys.net/17/49/2010/npg-17-49-2010.html http://www.nonlin-processes-geophys.net/17/49/2010/npg-17-49-2010.pdf 49 63 2010-02-02 A study of the phase instability of quasi-geostrophic Rossby waves on the infinite β-plane to zonal flow perturbations L. Marié louis.marie@ifremer.fr Laboratoire de Physique des Océans, UMR6523 CNRS/IFREMER/IRD/UBO, IFREMER/Centre de Brest, 29280 Plouzané, France The problem of the linear instability of quasi-geostrophic Rossby waves to zonal flow perturbations is investigated on an infinite β-plane using a phase dynamics formalism. Equations governing the coupled evolutions of a zonal velocity perturbation and phase and amplitude perturbations of a finite-amplitude wave are obtained. The analysis is valid in the limit of infinitesimal, zonally invariant perturbation components, varying slowly in the meridional direction and with respect to time. In the case of a slow sinusoidal meridional variation of the perturbation components, analytical expressions for the perturbation growth rates are obtained, which are checked against numerical codes based on standard Floquet theory. Berloff, P., Kamenkovich, I., and Pedlosky, J.: A mechanism of formation of multiple zonal jets in the oceans, J. Fluid Mech., 628, 395–425, 2009. Dritschel, D G. and McIntyre, M E.: Multiple jets as PV staircases: the Phillips effect and the resilience of eddy-transport barriers, J. Atmos. Sci., 65, 855–874, 2008. Galperin, B H., Nakano, H., Huang, H., and Sukoriansky, S.: The ubiquitous zonal jets in the atmospheres of giant planets and Earth's oceans, Geophys. Res. Lett., 21, L13303, doi:10.1029/2004GL01969, 2004. Gill, A E.: The stability of planetary waves on an infinite β-plane, Geophys. Astro. Fluid, 6, 29–47, 1974. Lee, Y. and Smith, L M.: Stability of Rossby waves in the β-plane approximation, Physica D, 179, 53–91, 2003. Lorenz, E N.: Barotropic instability of Rossby wave motion, J. Atmos. Sci., 29, 258–264, 1972. Manfroi, A J. and Young, W R.: Slow evolution of zonal jets on the β-plane, J. Atmos. Sci., 56, 784–800, 1999. Manfroi, A J. and Young, W R.: Stability of β-plane Kolmogorov flow, Physica D, 162, 208–232, 2002. Maximenko, N A., Melnichenko, O V., Pearn, N P., and Hideharu, S.: Stationary mesoscale jet-like features in the ocean, Geophys. Res. Lett., 35, L08603, doi:10.1029/2008GL033267, 2008. Nakano, H. and Hasumi, H.: A series of zonal jets embedded in the broad zonal flows in the Pacific obtained in eddy-permitting ocean general circulation models, J. Phys. Oceanogr., 35, 474–488, 2005. Panetta, L.: Zonal jets in wide baroclinically unstable regions: Persistence and scale selection, J. Atmos. Sci., 50, 2073–2106, 1993. Rhines, P B.: The dynamics of unsteady currents, chap 7, 189–318, John Wiley and Sons, Inc., New York, 1977. Rhines, P B.: Waves and turbulence on a beta-plane, J. Fluid Mech., 69, 417–443, 1975. Sivashinsky, G I.: Weak turbulence in periodic flows, Physica D, 17, 243–255, 1985. Thomson, A. and Young, W R.: Baroclinic eddy heat fluxes: zonal flows and energy balance, J. Atmos. Sci., 64, 3214–3231, 2007. Vallis, G K.: Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, Cambridge, UK, 2006. Vallis, G K. and Maltrud, M.: Generation of mean flows and jets on a beta plane and over topography, J. Phys. Oceanogr., 23, 1346–1362, 1993. Young, W R. and Rhines, P B.: Rossby wave action, enstrophy and energy in forced mean flows, Geophys. Astro. Fluid, 15, 39–52, 1980.