Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 16 4 2009 10.5194/npg-16-569-2009 http://www.nonlin-processes-geophys.net/16/569/2009/ http://www.nonlin-processes-geophys.net/16/569/2009/npg-16-569-2009.html http://www.nonlin-processes-geophys.net/16/569/2009/npg-16-569-2009.pdf 569 577 2009-08-31 Large-scale instability of a generalized turbulent Kolmogorov flow B. Legras legras@lmd.ens.fr B. Villone Laboratoire de Méteorologie Dynamique, Ecole Normale Supérieure and CNRS (UMR8539), 24 rue Lhomond, 75231 Paris Cedex 05, France Istituto di Fisica dello Spazio Interplanetario di Torino, INAF, 4 c. Fiume, 10133 Torino, Italy We present an analytical study of the large scale instability of a generalized turbulent Kolmogorov flow, i.e. a periodic shear flow where the molecular viscosity has been substituted by an eddy viscosity parameterized with the Clark-Smagorinsky model and where the external forcing is adapted to maintain the flow against this dissipation. We employ multiscaling technique assuming a scale separation between the basic scale of such a generalized <i>turbulent</i> Kolmogorov flow and the largest scales of the flow. The main result is that an amplitude equation for the large-scale secondary flow is obtained which exhibits, like for the standard Kolmogorov flow, an instability of the negative viscosity type. We find that the presence of mirror symmetry in the basic flow is a necessary condition and that further propagative and nonlinear contribution are produced otherwise. 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